DROBOVOLSKY, M. V. Liquid fuel rocket engines - design fundamentals. Mechanical Engineering Publishing House, Moscow, 1968.html

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FTD-MT-24-135-69-Vol I of IL

FOREIGN TECHNOLOGY DIVISION

AD695929

LIQUID FUEL ROCKET ENGINES — DESIGN FUNDAMENTALS

Wistribution of this document Is unlimited. It may be released to the Clearinghouse, Department of Commerce, for sale to the general oublic.

Reproduced by the CLEARINGHOUSE for Federal Scientific & Technical Information Springfield Va. 22151

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FID-MIT- 24-135-69-Vol 1 of II

EDITED MACHINE TRANSLATION

LIQUID FUEL ROCKET ENGINES — DESIGN FUNDAMENTALS By: M. V. Dobrovol'skiy

English pages: Cover to 301

Source: Zhidkostnyye Raketnyye Dvigateli. Osnovy Proyektirovaniya Moskva,

: . '

Izdatel'stvo "Mashinostroyeniye,'

1968, pp. 1-396.

THIS TRANSLATION IS 4 REMDIVION OF THE ORIGL- MAL FOREIGN TEXT WITHOUT ANY 4MAL YTICAL OF

EDITORIAL COMMENT. STATEMENTS OR THEORIES PREPARED BY, ADVOCATED OR MPL/ED ARE THOSE OF THE SOURCE

AND DO MOT WECESSARILY REPLECT THE POSITION TRAMSLATION ФГУ:

OR OPINION OF THE FOREIGN TECHMOLOGY 0+ POREIGH TECHNOLOGY Division У. УР-АРВ, ОМЮ.

ЕТО-МТ- 24-135-69-Vol I of II Dew 27 June 19 69

C1-ACCESHOM MO. 98-DOCUNENT Loc

TM9500997

@.T1T

"6 LIQUID FUEL ROCKET ENGINES-DESIGN FUNDAMENTALS

A LTE NMEA пиво есче ити < | DATA HANDLING PAGE

TOPIC Tacs

spacecraft propulsion equipment, liquid propellant engine, nozzle flow, combustion chamber, liquid rocket fuel, combustion kinetics, fuel injector

47-SUBJECT AREA

21

9 -АиТмОй/Со-лиТнОй$

DOBROVOL'SKIY, M. V,

1O-DATE OF INFO

43-SOURCE

ZHIDKOSTNYYE RAKETNYYE DVIGATELLI. OSNOVY PROYEKTIROVANIYA (RUSSIAN) 60401 63-SECURITY AND DOUNGRADING | HFORMATI On 64-CONTROL MARKINGS |97-HEADER C\ ain UNCL, O NONE UNCL 7e-REEL FRAME WO. |77-SUPERSEDES 78-CHANGES -eeoenapmi ca WO OF PAGES a 1889 0850 UR 615 CONTRACT NO. x REP Acc. MO. PUGLI SMING DATE TyPa PRODUCT REVISION PREG 65- 94-00 TRANSLATION NOME he но 02-UR/0000 68/000/000/0001/0396 ——"��—

АВУТИАСТ

given.

fuels and materials and t foreign and domestic public methodical and do not pertain book is a textbook for students o and can also be useful to engineers

wee

engines (ZhRD). chamber and the propulsion system on the whole, of the theory, methods of calculation and description of sub- assemblies and «nits of devices with liquid fuel rocket engine are Processes of expansion of gases in nozzles, carburetion

and heat exchange are expounded, as well methods of profiling nozzles, calculation of injectors, determination of forms and volumes of

An analysis of work of installations with Supply systems with turbopump

combustion chambers, h open and closed circuits is given, assemblies ard pressure feed systems with gas, powder, and liquid pressure generators are considered, work of installations with closed circuits and methods of power coordination of such circuits, were given, in particular, a description of the propulsion system of the rocket "Vostok."/{/The book gives actual data on structures, r characteristics on the basis of

Examples of designs are purely

ons,

(U) Thie book presents design fundamentals of liquid fuel rocket It considers questions of design of the engine Basic statements

Much attention is allotted to

Examples of propulsion systems

any defined type of engine. igher educational institutions graduate students specializ-

The

TABLE OF CONTENTS

U. S. Board on Geographic Names Transliteration System

хофоефоооео Vv

зооооооофооо ооо овоо vi

Designations of the Trigonometric Functions

List Of ACTONYMS se eee sees eeeceeceeeenseeccce,

OCH ee men eter neee vil

Preface........,

ie в 1х

Chapter I. General Remarks

a 1

В. Classification of СН�" Fuel.

ca rn 1

1.2. Basic Parameters of ZhRD

a 12

1.3. Systems of Loss Coefficients, Expenditure of Fuel and Basic Dimensions of Nozzles

i ee a 20

Chapter II. Nozzles of a Liquid Fuel Rocket Engine............ 26 2.1. Types of Nozzles and Basic Requirements of Them..., 26

2.2. Losses in the Nozzle of a Liquid Fuel Rocket Engine.........

a 29

2.3. Designing of Conical NOZZLES eee cece cece eee 42

2.4. Basic Initial Positions in the Construction of a Profiled №0221е........,....

оооооооров ооо ооо оовово 43

2.5. Shortened and Optimum Nozzles

ооо ко овововне 48

2.6. Approximate Method of Construction of a Contour of an Optimum №221е.,......

ооо ооо ооо ооо о вос 24

2.7. Operation of the Nozzle in Off-Design Conditions

with High Counterpressure,

he ere 60

2.8. Work and Characteristic of Nozzles with a Central Воду...............

a 64

i 2.9. Calculation of Nozzles with a Central Воду......... 73

Chapter III. Carburetor and Chamber Heads of ZARD.......0..64. 86

3.1. Basic Elements of Carburetion and Combustion Processes

иво н иво во ооюоооввоюонь.. = 86

D2 Spray INJOCEOTS. eee eee ee eseeeeees ease cn, 102

—— ЧЩ

3.3. Centrifugal ИИ 112

2.4. Тио-Сотропеп+ Injectors

i rn 132

FTD-MT-24-135-69 1 ad ila

|

3.5. Chamber Heads for %Пе 218О.........ъъьзееооновеоов о

3.6. Influence of the Hea’ on Carburetion and Specific

АЦ. с. а oe cc ees ce ene ee we etele ceclls po cece.

Chapter IV. Cooling of CORD. ccccccvcccvevccsvtsescevessesevece

4.1 Heat Exchange in ZNORD,. . crcvcovcvcccvccccvsceseceeees

4.2, Methods of Cooling the ZNRD.....cceeccccsevcencvene

4.3. Process of Convection Heat Transfer from the Gas to

the ШИ со ооо oleic Ио в о ИИ, ИИ И...

4.4. Internal Reiations of Energy and Impulses for the

Boundary Тауег......озовюововоь ооо обо воен совоноосо

4.5. Solution cf the Integral Relation of Energy.......

4.6. Calculation of Convection Heat Exchange in a

РОВ нь Зе о оо оба о еее о о бед о ве В «о вла в

4.7. Conversion of Convection Thermal FlowS...cscecceees 4.8. Determination of Radiant Heat FlowS....cccceccscese

4.9, Determination of Heat Transfer from the Wall to the Cooling Liquid......

4,40, Forms of Coolant Passages of Chambers of a

ПЕ, оо Мер т Зе ое те РВ. ЕСМ. ве we ae

4.11. Calculation of Heat Transfer in a Ribbed Coolant.

Passa Зсезевоео ооо оо ооо ово ово ооо оо воре ооо воосово

4.12. Calculation of Cooling of a ZNRD, ооо ооо оо ооо 4.43. Certain Special Cases of Cooling ZNRD... cece eee eee Chapter V. Chambers of Liquid Propellant Rocket Engines.......

в... Shapes and Examples of Existing Chambers of Liquid

Propellant Rocket ENZineS...cocccccvcccvevrscvscesene

5.2. Determining the Volume of the Combustion Chamber...

5.9. Unstable Burning. .cccscvcccccccvccvsvecesvescescesese

5.4. Start and Shutdown of Engine... ..cscccccsesvecececs

5.5. Aftereffect Impulse......ccccsoes

5.6. Engine Chamber Strength Calculation.......

eeoeeeseeeeoeeeeoesveen

5.7. Additional Ветагк$.......ъьзовоовоооооововеовосово

FTD-MT-24-135-69

11

137

Lis, 165 165 174

137

196 212

220 225 238

248

302 310 214 221 329 336 342

Chapter УТ, Propulsion SYStOMS cece eee c eee ceccecaesececevce, 345

6.1. Feed И 345

6.2. Propulsion Systems with a Turbopump Feeding SUPERS ани ое аа, 348

6.3. Thrust and Specific Thrust of a Propulsion А ЗУМ ии ие рь и иконе озненианинннь 358

6.4. Fuel TANKS .. cc ceccceccecee.

оооооовово ооо ооо ооо вое 361

6.5. Feed System EQUIPMENt eee eck eee eeeeecees cece, 370

6.6. Determination of Feed Pressure and Hydraulic } Characteristics of the Feed SYStOM. . ccc eecccccecee. 381

6.7. Systems of Control and Adjustment of a ZORD........ 388

6.8. Examples of Systems Carried Out with Turbopump | Feed

FETS O08 SMe eithe ° Sisie Wes 0 069 0% Fis e Org ons 9 ohare tL 397

Chapter VII, Turbopump 9168... оон рооонозавичень, 412 7.1. Pumps for Feeding Components to BARD... . ewbev cee ia 413

7.2. Calculation and Characteristics of ZhRD Pumps....., 433 7.3. TNA Turbines

ооо ооо носов о оо о ооо вево ооо 446

7.4, Joint Operation of Turbine and 0) ньньь 465

7.5. Gas Generators

i rn 479

Chapter VIII. Closed-Circuit Propulsion SYSCEMS ..... eee eee ees 496 8.1. "Gas + Liquid" Closed CLVCULE. eee eee cece ee eueee 498

8.2. Closed "Gas + Liquid" Circuit without 2160......... 544

8.3. Closed "Gas + Gas" Circuit

ооо во оо ооо воово сос ое 520

Chapter Ix, Propulsion Systems with a Pressurized Feed еде,

ооо ооо ооо бов ооо ооо ово ооо 525

h 9.1. Gas Cylinder Feed SYStCM ине ююневивань, 527

9.2. Ехатр1ез ог Propulsion Systems with Gas- ‘ Pressurized Feed...

re ee re 537

9.3. Reduction Valves of Gas PrESSUTE... ии... ,., 547 9.4, Characteristics of Reduction VALVES... eee eee eeeee, 554

9.5. Design of Reducing Valve

ооо ово ооо ооо ово ооо 570 РТР -МТ-24-135-69 111

= ACL ea

9,6, Displacement Sup 7ТПАО, с. .озаоеье

9.7. Propulsion Syste Hybrid Engines [

Аррепаех. „ии иене нение ниве ee

Вал овгарпу нение ково ие вет

FTD MT-24-135-69

ply Systems with PAD and

ms with Preliminary Fueling and 149[, [150], [151]. ce vecceseneccers

iv

519

295 601

605

U. S. BOARD ON GEOGRAPHIC NAMES TRANSLITERATION SYSTEM

Block Italic Transliteration Block Italic Transliteration Aa Aa A, a РР Р р В, г

B 6 B 6 B, b С с С ес 8, в

В в Be У, у Тт T m т, ©

rer re G, Е ух yy U,u

Да Zé р, а o @ ФФ F, f Ee Ес уе, уе; Е, е X x Хх Kh, kh Жж MM 121, th Цц Ц у Ts, ts 8s 3. 2, $ Ч + Ч ч Ch, ch Hon И и I, i Шш Шш Sh, sh AA Aa У, у Щш ЦЩЦым Sheh, shch K & Кк К, К b > Sb 3 п

Ла Ла L, 1 Ыы HW ow У, у

Ми Ми М, м ьь ь ь '

Ни Hon М, п Дэ. ду Е, е

Оо 0e 0, о Ю»ю Ю» Yu, yu Па Пя Р, р Я я Aa Ya, ya

* te initially, after vowels, and after 4, b3 e@ elsewhere, en written as # in Russian, transliterate as y¥ or ¥, The use of diacritical marks is preferred, but such marks may be omitted when expediency dictates.

РТО-МТ-24-135-69 у

— bo oul = TS Янин а a. № едеимный ее — ет Г

FOLLOWING ARE THE CORRESPONDING RUSSIAN AND ENGLISH DESIGNATIONS OF THE TRIGONOMETRIC FUNCTIONS

Russian English ain sin cos cos tg tan ctg cot sec sec cosec csc sh sinh ch cosh th tanh cth coth ach sech csch esch ere sin 8111 arc cos cos~) arc tg tan71 arc ctg cot~) arc sec вес-1 arc cosec свс-1 are eh sinh71 are ch cosh™2 arc th tanh~) arc cth coth=2 arc ach sech™1 arc csch cach) rot curl lg log

| PTD=MT~24-135~-69 vi

о и

heey

List of Acronyms

Following is a list of conventional acronyms utilized in the

translation of this document.

ВАОР (БРДД) — long range ballistic missile EPK (SPH) — electropneumatic valve

GRD (TPA) — hybrid engine

Lif (English only) — lunar module

LPRE (English only) — liquid fuel rocket engine NDMG (HOMT) - nonsymmetric dimethylhydrazine PAD (MAX) — powder pressure accumulator

PGG (NTT) — steam-gas generator

RDTT (PATT) — solid propellant rocket notor SAS (CAC) — emergency rescue system

TNA (THA) — turbopump assembly

VRD (BPO) — jet engine

ZhAD (HAD) — liquid fuel pressure accumulator ZhGG (HIT) ~— liquid-gas generator

ZhRD (HPQ) — liquid fuel rocket engine

ZUR (3YP) — SAM

FTD=-MT-24-135-69 vii

la Bld a AS alii GAMES ARA ca A AR ei A an es me

ре т oR Pete AN AY

‘A, ЕАМ ma |

This book presents design fundamentals of liquid fuel rocket engines (ZhRD). We consider questions of design of the engine chamber and the propulsion system on the whole. Basic statements of the theory, methods of calculation and description of subassemblies and units of devices with liquid fuel rocket engine are given.

Processes of expansion of gases in nozzles, ecarburetion and heat exchange are expounded, as well methods of profiling nozzles, calculation of injectors, determination of forms and volumes of combustion chambers. An analysis of work of installations with open and closed circuits is given. Supply systems with turbopumo assemblies and pressure feed systems with gas, powder, and liquid pressure generators are considered.

Much attention is allotted to work of installations ‘with closed circuits and methods of power coordination of such circuits. Examples of propulsion systems were given, in particular, a description of the propulsion system of the rocket "Vostok."

The book gives actual data on structures, fuels and materials and their characteristics on the basis of foreign and domestic publica- tions. Examples of designs are purely methodical and do not pertain to any defined type of engine.

The book is a textbook for students of higher educational institutions and can also be useful to engineers and graduate students specializing in rocket technology.

FTD-MT-24-135-69 viii

PREFACE

More than ten years have passed since the first Soviet artificial earth satellite opened the era of conquest of outer space. The most important elements in space rocket systems are propulsion systems with liquid fuel rocket engines (ZhRD) which ensures only flights at earlier unattainable velocities within limits of earth's atmosphere, but also the possibility of flight in outer space. The apparent simplicity of ZhRD, the very idea of which was expressed by K. E. Tsiolkovskiy over 70 years ago, the creation of such engines has required knowledge and experience corresponding to the present level of science and technology, wide introduction

of methods of hydrodynamics, gas dynamics and heat exchange to engineering designs.

A contemporary propulsion system with a ZhRD constitutes a complicated system, the work of subassemblies and units of which 13 interconnected. Therefore, design of one or another unit is impossible to conduct separately, neglecting the construction and work of remaining elements of the installation, which creates definite difficulties in the presentation of corresponding material.

In the present textbook we have attempted a systematic account of design fundamentals of chambers of the engine and the propulsion system on the whole. By contents it is possible to divide the book into two parts: Chapters I-V, which expound basic questions of design of engine chambers, and Chapters VI-IX, which examine basic questions of design of the propulsion system on the whole.

FTD-MT-24-135-69 ix

<—s

врали jr twmmata cet ett ne nfo stearate Da Nara: ARERR rene

It is assumed that students in the present course are familiar with fundamentals of rocket technology and the theory of working processes in ZhRD. However, for convenience the first chapter presents briefly the basic ideas which are used in examining the various questions in designing 2180. For a best understanding of working processes and peculiarities of design of elements of these engines the basic design methods are illustrated by examples.

In view of the limited volume of this book, certain questions (turbopump assemblies, control, etc.), examined in special textbooks or aids are given in compressed form. Moreover, only the basic information necessary for a correct approach to designing installations on the whole are given. The author has tried to avoid mathematical computations in cases when they cannot be used for a direct calculation of the various elements of a propulsion system.

When writing the textbook we systematized information published in periodicals and books and also made use of earlier published works of the author.

We express our deep gratitude to Professors S. D. Grishin, F. L. Yakaytis, and Docent Yu. V. Krylov for valuable remarks and recommendations made in reviewing the book, and also Professor

G. B. Sinyarev for his valuable advice, given during a joint discussion of the book.

The author requests that readers send their opinions and critical remarks to the following address: Moskva, K-51, Petrovka, 24,

Izdatel'stvo "Mashinostroyeniye" [Mocksa, K-51, Merposxa, 24, издательство "Машиностроение ]."

РТО-МТ-24-135-69 х

CHAPTER f GENERAL REMARKS

The present chapter gives basic ideas and relationship between Parameters which must be known for Studying desig¢n fundamentais of ZhRD. Moreover, it 1s assumed that the reader is acquainted with fundamentals of rocket technology and the theory of working Processes In chambers of ZhRD, in consequence of which ideas and relationships to be mentioned later are given in concise form without Proof. Those who are interested can find ereater detail on proofs and analysis of these relationships in works '25], [14], апа [2].

deals Classification of ZhRD Fuel

Liquid-propellant rocket engine is the name of rocket engines which use liquid fuel.

Liquid fuel and liquid oxidizer move from tanks to the engine chamber where, as а result of fuel combustion, high-temperature Gaseous products are formed (Fig. 1.1). In the nozzle they are expanded from the chamber pressure to the nozzle section pressure and flow into the Surrounding air at high speed. The outflow of

Gases from the nozzle is the cause of the reactive force of the engine,

PTD-¥T=24-1 35-69 1

Pig. 1.1. iagram and cycle of ZhRD.

Classification of ZhRD

The type of ZhRD is determined by some characteristic criterion (fuel, circuit, method of suprly, assignment, etc.). Figure 1.2 gives a diagram of classification of ZhRD in terms of basic charac- teristic criteria. Work of ZhRD with the different circuits, methods of supply, structural elements, conditions of operation and also basic properties and types of fuels used are explainea in subsequent sections of the book. Therefore we will not describe a given circuit in terms of noted criteria. Let us consider oniy the area of application of ZhRD.

Area of Application of ZhRD

The basic application of ZhRD is as the engine of rockets, whence the name liguid propellant rocket engines.

They are the basic type of engine of launch rockets of satellites or spaceships. Figure 1.3 shows the three-stage rocket "Vostok," which was used for orbiting the spaceship with astronaut Yu. A. Gagarin. Figure 4.29 shows the propulsion system of the first stage of "Vostok" liquid propellant rocket engine RD-107. Figure 6.7 shows liquid propellant rocket engine "Cosmos" RD-119 for the second stage of "Cosmos" rockets launch. Figure 1.4 shows a dtugram of the three-stage launch rocket "Saturn-V" with the "Apollo" intended for flight to the moon. Thrust of the propulsion svstem ot the first stage is 3400 t (13.4 MN). Figure 6.32 gives an ansembiy diagram of the propulsion system of the first stage of vne launch rocket.

PYD-M 2-241 35-69 2

pee este

oxyzen

Py Nitric acid oxidizer Hydrogen peroxide ] Iwo- a Fluoride, - Component Ее. ГАНуа горит peroxide | courant Няни Е eto. With open ву cireuit | cireuit With closed — By means circuit apor from у > -| woeling systen | Ж = medium apping gaa roine pump haaber р ky supply Г etc. _ ayst oe Ц Displacement aa A `

Ballistic rocket

» Weather rocket

By operational | conditions

Ry construction of separate elements

Fig. 1.2. Classification of ZhRD.

FTD-MT=24-135-69 3

[Pultichamber — |

Ц замы зы

Fig. 1.3. "Vostok."

5 /

fe" ‘

Liquid fuel rocket engines are widely used in long-range ballistic missiles (BRDD) and those of medium range of operation, ABM, SAM, and 2110 in metecrological rockets. Figure 6.31 shows the assembiy 3° a-rarm of the propulsion system of the "Atlas" BRDD.

Liquid fue! rocket engines are also one of the basic types of chgine utilized in spaceships for braking and in orientation system (Fig. 1.4). Fisure 9.6 shows a diagram of an installation for vorreci ing tue speed of a spaceship.

an tars tm si actigiin ined

Fig. 1.4. Diagram of "Saturn" launch rocket with "Apollo" and emergency rescue system (SAS): 1 = first Stage; 2 - adapter becween first and second stages; 3 — second Stage; 4 — adapter between second and third stages; 5 — third stage; 6 — instrument Section; 7 - LM; 8 — engine section; 9 — "Apollo" command module; 10 — SAS; 1l — crew section; 12 — adapter between launch rocket and Spaceship.

Besides use in rocket Systems, ZhRD have found application as the engine of other than rocket systems. Liquid fuel rocket engines are set on aircraft as a basic engine as well as for start boosters

(Fig. 1.6). They are also known to be used as the engine of torpedoes.

8 Ge

Fig. 1.5. ZhRD for orientation system: a) 90 kg {883 N) thrust; b) 7.2 ke (70.6 N) thrust.

Fig. 1.6. Kbei-Kns ai

Jn contrast to us fuel, in Z2hnRD

For @ ei

Fuel

the fuel is oxidizer

rerall atecavciauui.

ual thermal machines where fuel is actually

+ fuel.

yen oxidizer and fuel, properties cf the fuel are

ermined by their relationship, which is characterized by

excess Gxiduit ratio

ге , anv uo

У

oo Yo

— stoteniometrical and real ratio of expenditure

6

‘ sxiGises to expended combustibie respectively. Value a essentially

“sote the pasie propellant properties.

In ZhRD we distinguish fuel which is Spontaneously inflammable,

1.e., inflammable upon contact of oxidizer with fuel, and

non-self-igniting, 1.e., requiring an outside source of ignition.

We also distinguish two~=compenent and single-component (unitary ) fuels.

Liquid fuel rocket engines almost exclusively are bi-propellants. Monopropellants are used mainly to drive to turbopump assemblies (TNA) (in systems with PGG) and in certain low thrust engines [for example, engines of orientation Systems of spaceships (see Fig. 9.1)].

According to conditions of exploitation fuels are divided into high-boiling and low-boiling (cryogenic) fuels, the components of which under norma] conditions are liquified gases (for example, oxygen, hydrogen, fluorine).

Requirements on fuels of ZhRD can be divided into three groups: a) basic, b) Structural, c) operational.

Basic requirements are determined by the main problem — obtaining the greatest specific thrust with the smallest possible mass of the propulsion system. Finally they are formulated зо: the fuel should possess a large reserve of chemical energy and high density, and combustion products of the fuel should possess good thermodynamic Properties (value of as constant, isentrope index, etc.).

Structural and operational requirements are determined by the problem of Creating a propulsion System which is reliable, convenient in operation, and as cheap as possible. In accordance with these requirements one evaluates the physical properties of the fuel, cooling properties, ability to self-ignite and limits of inflammability, chemical stability, explosiveness, aggressiveness with respect to metals, toxicity, boiling point and fusion point, and, finally, cost of the fuel.

Thus, components of fuel have numerous and various requirements which Simultaneously can not be satisfied by one of the components,

although the possibility of use in fuels of almost all elements of the periodic system of Mendeleyev has been investigated [23].

In ZhRD the basic components of fuels are oxidizers on the basis of oxygen (pure oxygen or its compounds) and fuels on the basis of hydrogen and carbon (hydrocarbons, hydrazoic compounds, pure hydrogen). In the very near future it is possible to expect the use of fluorine and its compounds as oxidizer and compounds of boron, beryllium, and lithium as fuels.

Tables 1.1, 1.2, and 1.3 give certain basic properties of oxidizers, combustibles, and fuels.

Table 1.1. Basic physicochemical properties of certain oxidizers used in liquid fuel rocket

engines. i 1 full heat content % 3 | & : Я 016430 | | 8 . 1 < Е sf } 7 3 $ 2 ase i Фа: 2 м м 8 we |e $3. Кут, 11914 O, | 22 ид —3100 |—402 | 1140 | 54,3" и 0.1! Pluorine, liquid Fy —3000 1—3 |150 | 55.16] 85, 1613.2 №4716 2014 f —4140& ‚—2780 | 1510 |231 5/359, 16] 0,28 таг. Ozone | +3020 |+-2635 | 1700 | 21,761161,66] ~. Fluorine monooxride +2060 [+222 | 1820} 49 | 128, — water 18,01 ram —68370 |—15880) 1000 a a a

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11

1.2. Basic Parameters of ZhRD

Here and below the parameters related to the chamber of ZhRD will be designated in accordance with flow of the cycle of a ZhRD and characteristic sections (see Fig. 1.1). We will relate all parameters of a combustion chamber to section "2," 1.e., pressure in the combustion chamber, temperature, density, etc., will be

designated Po; Т›; Po, etc. In cases when we consider acceleration of gas in the chamber, we will use parameters of braked flow.

Thrust and Specific Thrust

Thrust and specific thrust are important parameters of ZhHRD. The formula for thrust traction assuming one-dimensional flow of gas through tne nozzle of ZhRD has the form

Ра р), (1.1)

where P — thrust; G — fuel consumption; P33 W33 Г} — pressure, speed, and area of cross section on nozzle section; p, — ambient pressure.

When Рз = Рн the engine works in design conditions so design thrust

Р.А С. (1.2) e If Рн © 0, 1.e., when the engine works in a vacuum, thrust in a vacuum РТО (1.3)

When P3 > P, or Pz > Py of the engine works in conditions of under-

expansion and overexpansion respectively.

Specific thrust, i.e., thrust referred to flow rate per second of fuel

р Ри

[уд -— specific]

12

or

Py m+ B(p,—p). (1.4) In design conditions Paw, (1.5)

Pint, (1.6) 8 where w= 0, +A (p,— д. (1.7)

ZhRD since on its value, in the end, depends the loading factor

of the rocket System (ratio of payload to Starting load) Figure 1.7 gives a calculation graph of change of initial mass of System for launching satellite with a mass of 5 io? & depending on specific thrust.

бы 7

Fig. 1.7. Influence Of specific thrust on initial mass of the System.

м КГМА

м № : ни ны у 2 Би чи ьм:

In heat engines one frequently meets the idea of Specific fuel

consumption, i.e., fuel consumption per unit of time for a thrust of 1 kgf.

Gp Pom me AF (1.8)

13

Inasmuch as specific consumption in ZhRD is simply connected with specific thrust, the concept of буд is rarely used.

Basic Relationships of ZhRD

The basic relationships characterizing work of ZhRD and defining its characteristics are: specific impulse 8, thrust coefficients K and K, and expansion ratio of the nozzle £3/fup: (Kp - critical].

Specific impulse 8 (or complex 8) of pressure in a combustion chamber 15

p= er (1.9)

If one were to express 6 as the ratio of fy p, vo mass flow rate,

then 8 has the dimension of speed ms. Therefore in Western literature specific impulse is usually called characteristic speed and is designated by c*.

The theoretical value of 6 is calculated by the expression obtained from the equation of consumption

pa BE (1.10)

Неге

аут, (1.11)

n — polytropic index of expansion of combustion products from Po to

Pxp .

Values “RAT, and AL depend on the kind of fuel and barely depend on other parameters of work of an engine (within limits of 1-2%). Therefore we approximately consider that theoretically ® depends only on the kind of fuel and is a constant thermodynamic characteristic of a given fuel. For a given fuel the value of the 8 complex depends only on the qualitv of the flow of processes

14

in the combustion chamber and does not depend on processes in the с. Thus, for a given fuel & is a characteristic determining the

.

work of only the combustion chamber. Nozzle thrust coefficient K (or thrust coefficient )

К--Р.. (1.12) SapPr The thrust coefficient shows how many times engine thrust exceeds the basic component force of thrust fy. p.- Therefore, sometimes К ts ealled dimensionless thrust. The theoretical value of K is

‘alcuiated by the expression obtained from the formula of thrust

(1.13)

where mn. — mean index of Isentrope of expansion. It is more

convenlent to use the thrust coefficlent in a vacuum K :

K,=—., (1.14)

SupP2

When p, * 0 the calculation expresston for K is obtained from equation (1.13):

ae

fas т Г a and | =)“ у '- (=) eX x] 14 Sect Е

1 „1 Г ~ er (1.15)

le Pa _|

As we sce, K. does not depend either on work of the combustion chamber

or cn external conditions (ps) and is a characteristic determining

work of the nozzle of the chamber only.

In accordance with formulas (1.1) and (1.3) K and K.. are

connected by k=k,-£ ©, (1.16)

whence taking into account equality (1.12) we obtain the expression for determination of empiricai

К... = P= FaPa

(1.17; Л upP2 :

where P. — measured rea! thrust of engine. Comparing expressions (1.9), (1.12) and (1.14), we obtain the formula fcr determination P and P ГЕ

Py=3K; Py,,=3K,. (1.18)

Expansion ratio of the nozzie Г. (or simply broadening) is defined as the ratic

я (1.19)

| This value not only determines nozzle dimensions but also characterizes basic parameters of work of the nozzle p3/p3> М. (ог зрееа из) › B70 3- (Here and below the asterisks "#�"" designate parameters of braked flow).

(1.20)

16

”

me

А

еее, (1922) --у == 2, er 1-2) * |. (1.22)

"as tee (1.23) т = Ag —! 2, Е (1.24) Е т Е - (1.25)

Comparing resulting expression, we see that broadening of the nozzle is simply defined by any of the basic parameters of work

of a nozzle. Relation

ё-. 3 ‚ (1.26) Pe Е

is called the expansion area ratio of a nozzle. As we see from

а. (1.20), expansion ratio 6 for an assigned fuel, (.е.. _) and broadening of the nozzle Г: do not depend on a change of pressure in the combustion chamber.

Relationships (1.20)-(1.25) are obviously valid not only for @etermination of parameters on tne uozzie section, but also permit determining р%/р› M (or w), р%/р in any section of the nozzle with f= f/T.,..- Using the shown relationships, one can determine the change in p, w, р, T cver the iength of the nozzle. Figure i.8 gives a graph of the change of broadening depending on ratio p5/p and the isentrope of expansion n .

17

Fig. 1.8. Dependence f/f... =

(а...)

Characteristics of ZhRD

In ZhRD we distinguish two basie types of characteristics: throttle and altitude.

The throttle (or consumption) characteristic is the name of the dependence of thrust P or specific thrust Руд on consumption of eomponents G at constant altitude.

The altitude characteristic is the name of the change in thrust P or specifie thrust P_. depending on altitude of work cf the engine H (or an ambient pressure p_) during constant consumption.

The equation of the throttle characteristic P = f(G) is obtained trom the formula of thrust (1.1) converted taking into account relationships (1.9) and (1.26) and taking p, = p§:

Pu=G (2+ м (1.27)

fnasmuch as the sum in parentheses and member f 32; do not depend on ‘onsumption, the equation of the characteristic has the form of an tquntion of a straight line

P=AG-B.

‚ 1. 1.Ya are represented throttle characteristics plotted from equation (1.27) for an engine with different expansion ratio, and

18

ath a eth

as NAN

on Fie. 1.9b for an engine working on lana (P.. = Ро) ana ina vacuum {P. = 0). [Г there is a considerable decrease of

sonsumption as compared to eulculation conditions separation of flow from walis of the nozzle will occur (see section 2.7). т this case equation (1.27) will not be valid and the real characteris-—

tic will го through the origin of coordinates as shown by the dotted

го

Ly | е) рь=0 г

Siz. 1.9. Characteristics of 2580: а, b) throttle characteristic of thrust (P = f(G); с, 4) throttie characteristic of specific thrust (P = £(G))3; e, f) altitude

characteristic.

The throttle characteristic of change of specific thrust is obtained from formula (1.27):

— 3 1 Fs 94 ЛР т =. (1.28)

39

During work of the engine in a vacuum (P,, = 0) Patt 83, (1.29)

4.е., зрес1Г1с thrust in a vacuum does not depend on consumption.

On Fig. 1.9c, d are shown throttle characteristics P= f(3) and P.. = £(S) for engines with different expansion ratios and

улей

for engines working at various aititudes.

Dependences of thrust or specific thrust on flight altitudes (altitude characteristic) are determined directly by formulas (1.1) or (1.4), in which with a change of height only p,, 1s changed. From a comparison of these formulas it Is clear that the character-= istic of thrust and specifie thrust differ in altitude only by scale (Fig. 1.9f). The altitude characteristic of thrust or specific thrust in terms of aititude of work for engines with different

broadenings are shown In Fig. 1.9e.

Systems of Loss Coefficients. Expenditure of

$3. imensions of Nozzles

System of Coefficients for Evaluating the Quality of Processes in ZhRD

For evaluating the quality of flow of processes in ZhRD it is possible to use either efficiencies, which estimate the perfection of conversion of initial energy into effective work, or coefficients, which estimate the loss of specific thrust (impulse) due to a low- quality flow of processes of conversion of energy.

in the first case there will be the so-called power coefficients (efficiency), in the second, impulse coefficients.

In ZhRD the more commonly used are impuise coefficients, examined below. Power coefficients are detailed in [14] and [25].

If one were to designate ¢ on the loss factor of specific

thrust then specific real thrust could be defined as

- 30) Риа=Р'. (1.3

be

losses of specific thrust in general are determined by losses in

tA "5

the chamber, in the nozzle and on thermal resistance, which we will estimate accordingly loss factor in the chamber ¢,, nozzl2 coefficient

and loss faectcr on thermal resistance $ п? 30 that

P= FaPcPremr (1533)

Above we noted two basic parameters, determining processes in a combustion chamber and in a nozzle: complex & and K.. The distinc- al value of complex § , obtained on the basis of

a

experimenta ata by formula (1.9) from the value calculated by equation (1.10)

flow of processes in the combustion chamber, 1.e., losses in the

combustion chamber (for detail see section 3.6). Thus .-®. (1.32)

If, durinz a comparison of calculation and real values or complex 8, real expenditure is set equal to ealculation and the calculation of 8 is conducted at real value Pry» 3.е., Gi. = G and i =f , then

и

fie. 5;

' A Thus, $, characterizes the value of losses of pressure due to the low-quality of processes. Therefore ¢, is frequently called the

coefficient of fullness of pressure. ~

Placing in equation (1.32) the value of 8 from formula (1.10), we obtain

_WRTs

Фи Уят. *

21

whence, taking R.,.. = Ros

Та — (1.34) т. %

Formula (1.34) permits for known or assigned 6, approximately estimating the decrease in design temperature in the combustion

chamber due to losses in it.

Losses in the chamber are made up cf two forms of losses: losses on incompleteness of combusticn due +$с the low-quality flow of processes of conversion of fuel into combustion products (see section 3.1) these losses do not yield to exact calculation, are usually determined experimentally, and are characterized by the coefficient cf incompleteness of combustion 9. ‚г)з апа losses on rregularity of distribution over the section of the chamber of the ratio of components and specific weight flow, expressed by variation factor $.,- Thus,

es ‘л Ne

$ = Ferra (1.

Total losses in the combustion chamber are within limits Фи = 0.95- 0.99.

Losses in the nozzle are defined as the ratio of the real value of the thrust coefficient in a vacuum K, д defined on the basis of results of experimental data by а (4547), to its theoretical value calculated by formula (1.15):

a=. (1. 36) In general they are composed on losses on dispersion of flow

(Фрас )» on friction (9), inlet (¢.,)2 on nonequilibrium of the process of expansion (Gi. I» on nonadiabaticness (Фохл )› and also of losses connected with the presence of a boundary layer ($... уж)» and losses during expansion of a two-phase flow (op )- Thus,

Зе рес Р:р Pan teepTezatcy a2: (1.37)

22

REE

e.

Losses on thermal resistance take place only in a high-speed combustion chamber. Then calculation is given in works [2] and [110]. ic combustion chamber ¢.,.., = 1, and then expression

3? = ФТ». (1. 38 ) Determination of Fuel Consumption and Areas of Nozzle Sections f,, and РГ. =} 3 Let us examine now we determine taking losses into account,

fuel consumption ane areas of the critical and nozzle exit section

‘f and ГР.) for an assigned thrust known from thermal caiculation ma 2 и

of specific thrust P ana assigned or known loss factors 9,. and

i ,

с

Theoretical fuel consumption, i.e., neglecting losses

в“ (1.39)

ть

Real fuel consumption, i.e., taking losses into account, is necessary to ensure assigned thrust

И = >. (1.40)

From expressions (1.39) and (1.40) we obtain the relationship between real and theoretical expenditures of fuel:

G,=+, (1.41)

« [©

i.e., for production of assigned thrust it is necessary to pass more

fuel in order to compensate for losses.

Theoretical area of critical section f,,, is determined from к} Гогти1а (1.19)

G3 1 } =. (1.42) Real area cf critical section Са taking equations (1.32) and (1.41) into account ;

or Se aoe ae . (25643) р Ре

This means that the area of the critical section must be increased

Oniy in order to pass through it the additional fue} eonsumption

for compensating losses in the nozzle, this additional fuel eonsunption for compensation of losses in the chamber, not requiring an increase

in fe .

Let us define the area of nozzle section in a cut. According

tc the equation of expenditure theoretical Г and real f, wiil

accordingly be equal to

eee (1.44 Я: ) м. (1.45) Л» Жал: ть

Let us find the approximate connection between f to equation (1.30)

3 and f3- According

ит, (1.86)

On the basis of the equation of state

¥: RT, т; а: т (1.47)

24

en (Bhs reora(2)

then in accordance with equality (1.34) (1.48)

Putting in expression (1.45) the values G., 3: апа УЗ from 1

rs oO += a 64 м 2 ” fe tA — + . = ры w on funy . I> л ыы № 5 [97 — . i “J <

» taking into account

о 2 oi a be bus = . lm со <2 ‚ - © хх с ct А le >

бк _ Ine ah aha и. (1,39)

We see that the influen f icsse hows up on a sharper increase

in 73 вап 1т Г. . The cause of this is that the value of Рзи › р. :

besides the increase of expenditure, is influenced also by the

i decrease of real speed из as compared to w..

3

25

СНАРТЕР II NOZZLES OF A LIQUID FUEL ROCKET ENGIUIF

In the nozzle of the chamber of the en-ine there occurs expansion and acceleration of procucts of combustion, i.e., the transformation of thermal energy obtained in the combusticn chamber iuio kinetic energy of the motion of <ases. The quality and weight of tne whole propulsicn system depend on the quality of operation of tne nozzle, its economy and weizht.

2.1. Types of iiozzles and asic equirements of “her

At present there are used or there is investirated the possibility cf using; the following basic types of nozzles (Fiz. 2.1): conical, profilea and nozzles with a central bcdy.

nound nozzles

г)

с

.

Nozzles with a central body

d) e) Г) 5)

Fig. 2.1. Types of nozzles of the liquid-fuel rocket engine: a — conical; b -— profiled; ec — with an angular entrance; d ~— annular; e — with full external expansion; f - with partial inter- nal expansion; & ~- plate with free internal expansion.

Conical and Profiled 10221]ез

Conical nozzles have the supercritical section in the form of a cone with a direct generarix (lig. 2.la). They are the simplest to manufacture anu have been widely in rocket engines.

With respect to economy of the operation, 1.е., with respect to the inagnitude of losses, and weight characteristics, they yield to profiled nozzles and at present are almost completely displaced by them, finaing application only in certain engines of low thrust.

Profiled nozzles have the generarix of the supercritical section made in a curve, which coircldes with the streamline (Fig. 2.16, с). At present this is the most widespread type of nozzles of a liquid fuel rocket engine [ZhRD]. ‘There are nozzles with a smooth entrance into the supercritical section of the nozzle (Fig. 2.1b) and nozzles having а break of the reneratrix in the throat. The latter type of nozzle is called a nozzle with angular entrance into the supersonic section or, simply, nozzle with angular entrance (Fig. 2.1c).

Sometimes nozzles whose throat has the form of a circle, in contrast to nozzles with a central body, are called ordinary or

` 2

er ee

er at

в. —

round nozzles. But usually by conical or profiled nozzles round nozzles are implied.

Nozzles With a Central body

In recent years there has been intensively investigated thie possibility of the application in a 2Z2hRD of nozzles with a central body similar in principle to the type of nozzles successfully used in a jet engine [VRD].

Nere are the following types of nozzles with a central body.

Ring nozzles (Fig. 2.1d), the expansion of flow in which is limited by an annular channel with solid walls. In principle

the operation of annular nozzles does not differ from the operation of round nozzles.

jJozzles with full external expansion (Tig. 2.le), not having an external wall forming the flow after the throat. Frequently this type of nozzle Is simply called nozzle with a central body.

Nozzles with partial internal expansion (Fig. 2.1f) where the external wall determines expansion only up to a definite pressure.

Such type of nozzle is intermediate between nozzles shown on Fiz. 2.14 апа 2.1е. The application of these nozzles can appear expedient with the necessity of deep expansion and the acceleration of gas up to great M values.

Plate nozzles (see Fig. 2.1¢), thus called because of the plate form of the central body with a free internal surface of expansion, since after the critical section they do not have an internal wall.

Problem of the Designing of Nozzles and Requirements for Then.

From a thermal calculation of the engine there are known only dimensions of the throat of the nozzle fy, nozzle exit section

23

fy (or the pressure is assigned on nozzle P3)- In the designing

of the combustion chamber we also determine dimensions of the

entrance into the nozzle section. However, other important dimensions of the nozzle determining its form and design dimensions (in particular, the length of the nozzle and angles of inclination of walls of the nozzle in the entrance and outlet sections) are unknown to us.

The problem of designing the nozzle consists in the determination of such a contour of walls of the nozzle with which the following basic requirements for the nozzles would be satisfied.

1. The nozzle should have losses of thrust as small as

possible, i.e., as large a value of the coefficient of the nozzle Ф с 1s possible.

2. The surface of the walls of the nozzle at assigned fw and г should be the least, which decreases the weight of the nozzle and facilitates its cooling.

3. The construction and technology of manufacture of the nozzle should be as simple as possible.

As frequently happens in technology, the indicated requirements are to a certain degree contradictory, and the full satisfaction one of them leads to a certain impairment of other properties of the nozzle. Therefore, in the designing of the nozzle depending upon the assignment of the engine, we take a certain compromise solution.

2.2. Losses in the Nozzle of a Liquid Fuel Rocket Engine

Classification and Estimate of Losses

As was already indicated (see s@ction 1.3), losses in the nozzle of a liquid fuel rocket engine are estimated by the coefficient

29

Quantity Ф с depends on various kinds of losses in the nozzle, the basic of which are the following:

1. Losses to the dispersion of speed at the outlet of the п022]@ Фра.

2. Frictional losses of gas on walls of the по221е фр. 3. Losses at the entrance into the nozzle ax. 4, Losses to the nonequilibrium of the process of expansion Финер.

5. Losses connected with the nonadiabatic flow of products

of combustion along the nozzle. Sometimes they are called cooling losses Фолл.

6. Losses to the narrowing of the section of the flow Фе».

7. Losses taking place with the outflow of two-phase working substances @¢.

In certain cases losses in the nozzle are conditionally attributed to losses in thrust appearing due to the off-design nature of the operating conditions of the nozzle gap. This is inadmissible since losses of thrust due to the off-design nature of operating conditions do not depend on the quality of the flow of processes in the nozzle. However, sometimes in the comparative estimate of different contours of shortened nozzles, it is convenient to

_ consider these losses by introduction of the coefficient gq», referred to the nozzle.

Each of the indicated kinds of losses is estimated the appropriate coefficient @ (pac Gr,etc.) expressed as

30

P,— 4P,

= Р. (2.1)

упеге АР — decrease in thrust from a fiven type of loss. Knowin=

for every kind of los3 oy: we can determine %$с:

P

= oe ы 6 == ] — zr = в s -, =S('-$4)-w—n (2.2) =!

vaking Into account equation (2

tle

7 3

), WE will obtain

=> %—(n—1).

a (2.3)

It is nore convenient (and more usual) tc determine 4

*с not as the Sul $1 but as the product

Те FpecPrp Pen Paep PorsPeyn PQ: (2.4) The difference in the quantity 4

, determined by formulas (2.3) or (2.4) is small.

Accordins to results Of experinental d

ata, the total marpnitude ef losses in the nozzle is dete

riined by tie formula (1.36): Фе = Кал/Ка.

Let us examine the components of losses entering into expression (2.4),

Losses to the Dispersion of Орееч at the Outlet of the Nozsle

In deriving the equat

ion of thrus¢ ме с direction of the flow of ¢

onsidercd that the asses passing from the neazle is

a ie

i a

не.

in parallel to the axis of the nozzle. In reality, if in section fy the direction of the wall of the nozzle is not in parallel to the axis, then the velocity of the flow directed along the wall is deflected from the direction of the action of the tractive force (Fig. 2.2). The traction of the nozzle is determined only by the component of speed parallel to the axis Wee

Since w�<� then the thrust of the nozzle 4g smalier than the design thrust, which corresponds to speed w,. Let us determine the magnituce of losses to aispersion. With quite small values of в and with the ratio fify>3 it is possible to assume that in the nozzle exit section radial flow takes place, and the surface

DP; * const nas the for. of a sphere of padius р with surface Гу.

#16. 2.0. Determination of @pece

То determine the thrust let us Separate on surface fi the annular element with are 8 tncluucd between anples @ ana p+d6, and let us find tiie thrust of the clement. Tne axial component of the thrust of the element in a vacuum

ЧР... = =: we + P32nRB cos 8.

(2.5)

Considering, that

©, = ©, COs 3;

R mtn (2.6)

3108 Rak, sin’, ’

we fo

and suostitutine Cxpressions (2.4) treo eyuation (:.5), we wt cutain

€P ya on Bsn peont. d+ p,2n азс. ap.

Intesrating: this expression within llidis from o to Bay we obtrte wd

Pane (М ei sin cos B+ py sin сов В) ыы

% [= Geo) 4 пы (2.7) эм? В The flow rate threurh surface ft 6, 9 | чел. fr. зесоглапсе with equations (2.5) the surface of the elenent e .. г\ 5 = == — —ы ® (2.2 ве ла = 2лЮ aap = 2x ing. (een) and we flnally copy equation (2.10) in the for к, (2.12)

where @e — loss fictor of tiirust to the dispersion of flow in view of the nonporaullelism of tne outflow. ‘he dependence of

Фе ОП angie 8. is civen in Table 2.1. .}

Table 2.1

3s 0 4 8 | 12 № 20 2 | 2 | BY 40

24 fee | 1.00 b, отримали 0,980 ,97550.0:98 |

From the table it is clear thut for the nozzle with an openins

of 26, = 25° thrust PF will be 98.5% of the thrust determined by

2 э

34

the formula (1.3), and at angle 283 = 12° losses to dispersion will be a total of 0.3%. Although the civen deduction is correct only for the radial flow of vases over the nozzle, formula (2.11)

with a sufficient degree of accuracy permits estimating losses tn dispersion also for profiled nozzles.

For a decrease in losses to dispersion it is desirable te have angle bs as small as possible, and when 83 = 0 it will be Фр =1. However, as we will subsequently see (see section 2.5), in nozzles of rocket engines it is inexpedient to reduce anrle к 83 to zero, since with this frictional losses and weirht of the nozzle greatly increase. For the usually applied or conical

nozZles the angle at exit 28. is within limits 10-30°, and losses of thrust to uispersion are 0.3-1.5%.

Frictional Losses ir. the ’iozzle

As a result of the friction of gas on the walls, there appears a force acting on the wall of the nozzle in a direction opnosite to the tractive force ("pulling"” force). ‘he value of frictional force actin; on the annular secticn of the wall of the nozzle dF

1 (Fir. 2.3), ew? ЧР = Cy — cos 8-dF;, (2.13) where Py and м, — density and flow rate in the i-th section; Cry -

coefficient of skin friction.

35

<9. Go

The coefficient of skin friction Cos taking into account the compressibility and heat exchange, can be calculated on the basis of the theory of the boundary layer. Approximate values of Cp can be obtained by using the semi-empirical formula

Суб» (Не му, (2.14)

where Cy, — coefficient of friction of the incompressible liquid,

r— recovery factor. For tentative calculations frictional losses can be taken as

20,80; C)_=0,003—0,006.

The total frictional force acting on the wall of the nozzle is equal to

= [4 (2.15)

The thrust in a vacuum, taking into account the action of frictional forces,

P,P — bP Wyo (2.16)

/

wnere @p -— frictional loss factor

Р.— АР. АР . og alae. (2.17)

For nozzles of a ZhRD values of the frictional loss factor lie within 0.98-0.995.

Example. Determine frictional losses in a nozzle and frictional loss factors @, and losses to dispersion @.. nerlecting the friction on the section of chamber prior to the throat.

36

mee en

The basic parameters of operation of the engine and dinensions of the nozzle

А-ТЮК 5/2 487 му/а?); 6125 Кв/ез Ти=З0МР К: я. >12;

R=WS kefow/ka-deg (337 S/upden); Rup=30 am: Prx=Pa=0,7 keF /en2 (0,069 HN/m?): R= 1038 an; Бью; ©.

тре contour of the nozzle is shown in Fie. 2.4.

Fir. 2.4. To sictermiination АР.; Civisisn on seetions.

Solution. ‘Yo Getermine frietionnl iosses let us divide thre evions and uetemnine frictional

losses according te parameters referred tc the middle of the section.

The coefficient of friction is jietcruined anproxinmetely b:

formula (2.14).

Let us consider $г = 0.903 and r = 9.0%. 3: Кпом1иа the nean values of ff let us vietermine quantities pim.Mewe by the well-kneown relationships of газ dynamtes. Ort8 of the ecalculstion

are civen in Table 2.2. Tne total fristiensli feree: a ВР Фарит кзг(ЗИ Н). =? Е

Let us define 1055 Гасфогз фр агО Фе. Phe thrust in a vacuun

P= G ло + 3.38-10-2.0,7-10% „= 3487 kof (34 200N).

The frictional loss factor

Pa— bP.

ty"

The loss factor to dispersion

31,7

e Фе 1 == = Anat on 0,0086. Table 2.2. Number ef sections 1 2 3 4 5 Angle of inelination of sontour f in deg. 30 21,5 16 il 4 cos 8 0,064 | 0,980 | 0,000 | 0,995 | 0,999 Averaze redive Rey 10 ine 37,5 38,5 4,5 91 102 Surface of the section 2 AF ORR -10 in 13,9 [13.0 [41,8 | 58,2 | 32,9 Sfep 1,56 1,65 4,62 | 9,20 / 11,6 РА 9,901 | 0,165 | 0.0386] 0.0140] 0.0104 mM? 0.176 | 3,50 | 7,40 | 10,36 | 11,3 й ome 0,6 (+ te rm) 101 [16 |115 |143 | 1.47 Cy-108 0,287 | 0,258 | 0,227 | 0,21 | 0,204 91.10% в2/о2 2.9 314 517 617 643 @ in kat -92/u4 0,634 | 0,154 | 0,0432 | 0,0197 | 0,0158 За ай 6.2 | 1:51 | 0,4% | 0.193 | 015 8 = -10—4 663 |242 [11,2 | 6,08 | 5,08 c, = 10 | em | ose | ans | 104 Prietional . foree on the section in kef 2,38 7,95 10,5 7,4 3,41 АРС, = AP, cosp'in x 123.4 | 78.0 103 [72,6 | 33,4

38

Losses at the kntrance Into the Nozzle

With gas flow in the Suberiticas part of the nezzle with a steep turn of Peripheral flow streams in a direction toward the axis of the nozzle, compression of Gas streams flowing near the axis of the nozzle occurs. As a result of this the pressure in the eentral Streams is set higner than the pressure around the wall of the nozzle. Fig. 2.5a shows a diagram of the distribution of pressure in the section I-I. In conformity with such a distri- bution of pressures along the section in a peripheral region the flow rate is set hicher than that near the axis. Therefore, at the entrance into the region of the throat the peripheral flow accelerates earlier up to the critical speed than does the flow near the axis. The surface of the critical speed АО 15 deformed, obtaining a convex form. With Great curvature of the entrance section the formation of compression Shocks and also the Separation of flow is possible, Furthermore, since the exact form of the Surface of the critical Speed is unknown, losses can appear due to the noncorrespondence

of the real Character of the flow ir. the entrance part of accepted desicn schene.

a) i? b)

Fig. 2.5. oXplanation of entrance losses: a) appearance of losses at the entrance; b) Vitoshinskty profile.

"as >)

All these losses are connected with the organization of the flow in the entratice part of the nozzle and therefore they are callea entrance losses in the nozzle. The magnitude of the losses is estimated by coefficient @s-—

The smooth entrance into the nozzle at which the surface of the stalling speed can be considered flat provides a contour of the subsonic part constructed оу the formula of Vitoshinskily (Fig. 2.5b):

R 1

eV SS (2.18) RIS (+3 cpp

Tne surface of the critical speed will be obtained quite close to

a flat surface if we profile the entrance part by an arc of a circle with a radius Rep=3Rsy. ‘The flat surface of the stalling speed is frequently taken as tne initial with calculation of the profile

of the supercritical part [43]. However, with such a construction

a relatively long entrance part of the nozzle is obtained, i.e.,

the dimensions and weight of the nozzle increase.

Experiments on the determination of the entrance losses in nozzles distinguished by a radius of curvature of the nozzle in the region of the criticai section showed that when Rea»>>0,65D. entrance losses are practically’absent. Therefore, in the profiling of nozzles of a ZhRD, depending upon the accepted design scheme, the entrance part of the nozzie in the region of the throat is

constructed on an are of a circle with the following radius of curvature [38]:

Rexp= (0,65—1,5) Dry (2.19)

Entrance losses here are taken equal to zero, i.e., @s™!l.

40

eo

Other Losses

Losses to the nonequilibrium of the process of expansion (@ 9). With very great broadenings, when the thermodynamic temperature of products of combustion in the nozzle decreases down to values of 1000-1700°K, or with the reduction of the time of stay of gases in the nozzle (for example, in microengines), the process of expansion of cases can flow partially or completely unbalanced (in detail see work [25]). With this the magnitude of specific thrust drops as compared to the specific thrust determined with equilibrium expansion. Losses to nonequilibrium can reach 5-104. Quantity ep can be determined by estimating the degree of nonequilibriun.

wosses due to the nondiabatic nature of the process of expansion (Фозл). The cause of these losser (also called cooling losses) is the heat removal from the flow into the wall. With this if the heat is reuoved irrevocably (for example, with cooling by a special component not utilized as fuel, or in the case of uncooled chamber) the specific thrust decreases as compared to specific thrust determined with adiabatic expansion. The magnitude of these losses with intense cooling can reach 3-5%. If, however, there is used regenerative cooling, at which the heat, tapped by the coolant, returns back into the chamber, then there will be no losses of specific thrust due to heat removal from the flow. Conversely, there will occur a certain increase in p,, since in the nozzle heat from the flow is removed at lower pressure than the pressure at which this heat returns into the chamber. However, an increase in Py, here is a fraction of a percent and therefore in calculations is not considered. Calculation of the influence of the nonadiabatic State on Py, is examined in work [2].

Losses to the narrowing of the flow owing the boundary layer (@eym). Due to the presence of tte boundary layer, and also the formation at the wall of a special wall layer, for internal cooling real areas of the cross section decrease and the character of flow in the nozzle

NY

is distorted, which in the end can lead to a loss in speed and thrust. The magnitude of these losses is small and is considered in the total estimate с.

Losses with expansion of two-phase flow (¢). In the use of certain forms of high-calorie fuels (for example, metallized fuels) in products of combustion small particles (for example, condensed oxides) can be formed. In this case in the nozzle there occurs expansion of not a uniform but a two-phase gas flow (heterogeneous). With the expansion in the nozzle the drop in temperature and increase in velocity of the particles attracted by the gas rlow occur more slowly as compared to the change in corresponding parameters of the gas. This leaus to a lowering of the specific thrust. Losses in specific thrust with expansion in the nozzle of a two-phase flew can reach considerable values, of the order of 3-10 and more percents, 2nd sharply reduce the effectiveness of the

application of high-calorie fueis. 2.3. Desirning of Conical Nozzles

Considering, the dependence of losses at the entrance, to

dispersion and to friction on form and dimensions of the nozzle,

in the designing of conical nouvzies it is possible to recommend

the following values of the basic geometric dimensions of the

nozzle (Fig. 2.6). ‘he entrance part is designed from the condition of providing the absence of entrance losses.

‘Pig. 2.6. Designing of conical nozzles.

The angle of entrance into the nozzle part

Зв»: =45--807. (2.20)

The radius of curvature of the eritical part of the nozzle Rem = (0,65— 1 ) Duy.

The radius of curvature of the entrance part of the nozzle

Ri= (0,35—0,5) Dp. (2.21)

The ancle of the outlet part of tac nozzle 28, 45 selected on the basis of experimental data concernine the Agpendened of the product Фра р ОП the aperture ansle of the supercritical part cf the nozzle eb. A typleal rraph of suen а vependence {is shown on

Pie. 2.7. Lsually

2B, = 25—30°, (2,20

A certain impairnent cof GUaNntity ¢pacGrp when 28, = 30° ts compensated by a decrease in lencth and, consequently, weight of the nozzle,

The main ueficiency in conical nozzles is that even with the most acceptable angles 28; the nozzle is obtained relatively lonc, heavy anu nas comparatively great losses to dispersion and friction.

vite

T Fic. 2.7. Dependence of €pacX Xen OF Angle 28.. э

Qs

0. 20 “M2;

Profiling the contour o¢ the nozzle permits making it shorter and lignter with 3 simultaneous decrease in losses.

2.4, Basie Initial Positions in the Construction of a Profiled ffozzle

Let us examine the basic positions from which one proceeds

>

‘in the construction of? the contour of the Superrconiec nart of a

profiled nozzle.

No

An ideal nozzle is called a nozzle with isentropic flow and a uniform fiow of gas in nozzle section parallel to its axis. Thus, for an ideal nozzle losses to dispersion are equal to zero. As we will further see, such 2 nozzle is not the best for a ZhRD. However, the contour of an idea} nozzle is usually initial for

obtaining more rational contours of profiled nozzles in a ZhRD. т.

The positions which are the basis in the design of ideal nozzles are the initial ones also in the designing of other types of . profiled nozzies.

The basis of methods of profiling the supersonic part of the nozzle is the method of characteristics. In the supersonic part of the nozzie the fliow of causes is expanded and accelerated, passing through an infinitely large quantity of rarefaction waves (Haen lines or characteristics). With the passage of flow through the rarefaction wave the direction of streamlines of the flow of gases is also changed.

The problem of the profiiing of a nozzle consists in the construction of such a contour of the nozzle at which the gas accelerates up to 2 set speed without the formation of shock waves, and the motion of tne flow is directed either at a definite set angle 83» or in parallel to the axis of the nozzle.

Along the length of the longitudinal section of the nozzle it is possible to scparate the following basic sections characterizing

processes in the nozzle (Fig. 2.8).

on

Fig. 2.8. Basic regions

of the process of expansion of products of combustion in the nozzle.

we

yy

I. Region cn the left of surface AOA' — region of subsonic gas flow. in it the acceleration of the subsonic cas flow up to a stallinre speed occurs. The contour of the subsonic part of the nozzle is profiled proceeding from the condition of providing the absence of entrance losses in the nozzle (see section 2.2).

II. ‘Surface AOA' — surface on which the speed of gas reaches | the speed of sound at a given point, 1.е., the surface of the stalling speed. The form of it in general can be hoth flat and convex und is uevtermined by the contour of a subsonic part of the nozzle. As yet there is no accurate solution to problem of the determination of the form of the curface of stalling speed. With different methods of profiline the nozzle we proceed from different foras of the surface AOA'. ‘he most widespread assumptions are

фл ль

about its flat or spheric forn.

é

III. kegion AA,0,4,/A’ — region of preliminary expansion.

a + ee eo `

Here the expansion and acceleration of gas occurs. On the section

of nozzle AA, there appears an infinite quantity of weal rarefaction waves A,C\; AxCy’... AnCe’, Which are characteristics. With transition through the rarefaction waves a decrease in pressure and acceleration of supersonic flow cecurs. In tim calculation of the contour cf

the nozzle the curvilinear section AA, is replaced оу a finite number of chords АА,; AjAz; A2Ay.... With this the broken line AA,... Aan

4s selected in such a manner so that the direction of the character- istics 4s changed by an infinite number (for example, 2°). Thus,

the infinite nusver of rarefaction waves of intinitesimal intensity is replaced by a series of rarefaction waves outcolnm from points

of the turn of the broken line. This replacement 1s equivalent to the assumption that expansion cecurs not continuously but at smali finite distances.

The angle of inclination of the velocity vector at the wall AA, With respect to the axils of tne nozzle continuously increases up to point Ane At point On the flow rate reaches the assiened speed.

IV. Region Ann — гег1оп оГ levelling off of the flow. In this region there occurs further acceleration of gas up to the assigned speed and levelling off of the direction of the motion of gas. The contour of the nozzle in this region is profiled fron the condition that at points Cys Cos Cys etc. reflection of rare- faction waves Aj'C,, AC; A/C, the direction of the wall coincides with the direction of the speed. ‘Therefore, at these points the angle of inclination of the wall A.C, is changed by an апг]е equal to the angle of rotation of the flow with passage throurh the rarefaction wave A,’C,. AC; AvC;,cv1c. The angle of inclination of the velocity vector of the Siow at the wall Anon from point An ur to point Ср continuously чесгсазез. With the attenuation of all rarefaction Ajy’C,, AyC,... wails of the nozzle will be peratlel to the axis of the nozzle (for on fuenl nousle). Thus, starting from point A, of the ungle of Inclination of the contour of the wall to the axis of the nowzle continuously decreases. Point hy is the inflection point.

On section AAD tic ancie of inclination of the contour continuously increases. Vith the construction of the prefile on this section, depenuda: apo tue Gtreanline according to which the profile of the nozzle is constructed, there can be obtained a larror or smaller length of tre ссвегагах hans and with the construction of the profile alonr: the Liniting streamline the section will reduce to a point. Resion IV will start directly from the critical section of the nozzle in which the contour of the nozzle will have а break (Fis. 2.9a). Therefore, sueh a nozzle is cellec a nozzle with an angular entry. Other things being equal, the length of the supersonic part of the nozzle with angular entry will be lcast. Therefore, in a ZnRD the application of nozzles with angular entry is expeaient.

У. Region C,0,C,’ — rcecion of uniform gas flow. For an ideal nozzle the direction of the motion of flow in this rezion is in parallel to the axis. ‘the flow rate is equal to that assifnec.

46

ne

Flo. 2.9. Different tnitial conditions for the construction of a crid of characteristics and profile of a nozzle: a) flat surface, anrulor entry ;

b) flat surface AO, smooth entry; c) m'm' -- calculated surface, roundee entry; а) radial flow up to surface m'm'; e€) radial flow up to surface ^.0, ; conelal entry.

To construct the rrid of characteristics, it is necessary to know the initial velocity distribution in some section of the Supersonic part of the noczle. If the fern and position of the Surface of the critical speea AOA' are known, then construction of characteristics can be conduetcd fron it. Пеге 16 153 песеззагу to consider that near the surface the sreed of flow insignificant.» exceeds the eritical speed, in consequenee of which the accuracy of the construction of characteristics 48 not freat. Therefore,

i бб x

sonetimes characteristics ire constructed startince from a certain section where.the [tow ulreiuy possesses Supersonic speed. Here the velocity distribution on this section is considered known.

With the profiline cf nousles af д énRD, several procedures of the construction of cheracteristics and profile have been used.

Frequently we proceed from the assumption that with a sufficiently Smooth profile of the subsonic part (large value of Rew.) , in the narrow section will be forned flat surface of the eritical speed,

17

Here the grid of the Cheracteristics will have the form shown on Fig. 2.9a, b. The profile of the nozzle shown on Fir. 2.9a has

an angular entry into the Supercritical part (nozzle with angular entry). The profile on Fig. 2.9b has a smooth entry. Here the length of the critical part of the nozzle will be greater than that

in the case of angular entry. In the construction of the entry e part, which provides a flat surface of critical speed for profiling the supercritical part of the nozzle both with anyular entry and °

in terms of the intermediate Streamline, 1t is possible to use tables of parameters of the supersonic flow of cases compiled hy О. Il. Katskova and Yu. D. Shnycglevskiy (6&4),

The characteristics can be constructed as 15 shown on Fir. 2.9с, 1.е., from the convex surface �'''�' in the region of the critical section [56] calculated by the Zauer method (42).

In certain works it is proposed to make the initial section of the supersonic part of the nozzle conical. Here the flow in the critical section actually is not examined anu it is taken that the supersonic flow starts from the Source at point K (Fig. 2.99, е). On the initial section the flow is taken radial and known either ud to section m'm' in the Supersonic part (Fig. 2.9d) or un to the characteristic Ann (Fig. 2.9e). Construction of characteristics Starts either from surface m'm' or from characteristic Anon» which is known in this case. However, an investigation of the field of flow in conical nozzles showed that the coincidence with the flow from the Source takes place Only on very distant sections of the Supersonic flow. Furthermore, the Supersonic part of the nozzle, constructed on the assumption of radial flow on its initial section, has a smooth entry and 1s obtained longer than for nozzles with an angle entry.

2.5. Shortened and Optimum Nozzles

Shortened Nozzles

Being based on the method of characteristics well-known from @as dynamics, we can profile an ideal nozzle with uniform and

це

Parallel flow on tie section.

If one were not to conside. frictional losses of thrust then such a nozzle will five the rreatest thrust. Let us consicer,

however, wicther the Application of ideal nozzles in a ZhRD is expedient.

The end section ЗС ог ап idea) nozzle, represented on Fir.

7.10, ives a very small increase in thrust, since on this Section

the cenerarix of the surface of the nozzle 15 almost parallel to

the axils. by shortening the nozzlc, it 1s possible to obtain

a considcravle cecrease in dimensions and wetrht of the nozzle,

wile ts very important for rocket Cricines.

Furthernore, it annears that cn the extrene section:

~© frictional loases of thrust exceed SNe Gnerense ir thrust on this section. Thus, with a definite

‘lecrease in lenrth of the Luend nozzle we cain not only in weicht Г § {

of the nozzle and dts alienstous out even in thrust. obtainea ty means of decreasin> tre senth of the are calleu siiortencd nozzles,

иные,

Such nozzles 117а]1 nozzle

Pir. 2.390. Comparison of an duct] nozzle with « chortened nozzle: 1 = thrust nerlecting frictional losses; 2 = frice- tional; 3 -— thrust taking, into account frictlonal iosses; 4 « tnrust of an ideal nozzle; 5 = thrust of a shortened nozzle; — contour of ar ideal nozzle.

ay

пои

a а

Optimization of the Nozzles

An ideal nozzle from which a shortened one is obtained is called an initial nozzle. It 1s obvious that for obtaining a shortened nozzle, which provides, for example, the assigned speed (1.е., having assigned fefe), the initial nozzle should be calculated at some large Mach number unknown to us. Therefore, it is necessary to construct a family of initial nozzles with different I! and to select from it an optimum shortened nozzle providing the assigned full With the least losses of thrust. It is possible also, having shortened, for example, all the initial nozzles of the siven family up to some assigned length lies to find the nozzle of assigned length giving the greatest thrust taking into account all losses.

It 1s possible, finally, to find the nozzle of the assicned surface nost advantageous in thrust, (i.e., in practice, ast ipned

weight), cutting off nozzles of equal surface from the initial nozzles.

Nozzles providing the greatest thrust of the propulsion system at definite assigned conditions ( fale length, surface, etc.), are called optimum nozzles. Obviously, in a ZhRD the most expedient is the application of optimum nozzles which provide, depending upon requirements for the installation, the greatest thrust at the assirned length or weight, or area fs, etc.

Fig. 2.1] gives a family of contours of initial nozzles with angular entry on which there are plotted curves of constant thrust coefficient in a vacuum which is determined into account rrictional losses. Such graphs are used for the analysis and selection of different types optimum nozzles. On them the contour of the optimum nozzle of assigned length 1s determined by the point of tangency of the vertical line, corresponding to the assigned length, to the curve of constant thrust. The contour of optimum nozzle of assigned surface will be determined by the point of tangency of the line of constant surface to the curve of constant thrust. On Fig. 2.11

50

Зав ала Га 2,

а $

it 1s clear that with the assigned thrust the contour of the nozgle of least surface. The contour of the optimum nozzle with assigned area fy will be determined by the point of tangency of the horizontal line corresponding to area fy with the curve of constant thrust.

The application of optimum nozzles gives high gain not only in thrust but also in aimensions of the nozzle. Pigure 2.12 gives a comparison of lengths and contours of different types of nozzles.

Fig. 2.12. Comparison of nozzles of different types: a) by contours; b) by lencths; 1 ~ ideal; 2 - conical; 3 — optimun; 4 — with a centre) body.

b) he

the optimum nozzle with the least length of the supersonic part will be obtained with the fulfillment of the contour with angular entry. For a ZhRD such @ nozzle is most profitable, but technological difficulties and the danger of flow Separation near the angular edge lead in a number of cases to the necessity of making the entry with small rounding.

In this case we found the optimum nozzle from a number of shortened ones obtained by cutting part of the initial nozzle. The contour of the optimum nozzle may also be obtained directly as a result of the solution of the variational problem on the finding of the contour of the nozzle of the greatest thrust under certain assigned conditions. Thus, in works C40}, [52], [56} the problen

52

wane

of determination of profile or the optinun aozzie 1s solved, which provides the greatest thrust at the assiyned lengch of the nozzle bes pressure in the combustion chanoer Po and ambient pressure

ges in work (50) the variationa) sroblen of finding the ontirun nozzle of least welght iz examined. In the incleated works

losses of thrust to dispersion ani to the off-desicn state of operating conditions of the nozzle ore considered; the remaining forms of losses (including frictional losses) nre not considered.

liaving, determined the optinun contour of the nozzle with the assicned length (or other parameter), we can construct a fanily of optimus nozzles and from this fanliy seleet the nozzle which Satisfies us the most with respect to other conditions (veirht, nidsection, frictiona] losses, etc.). Here frictional losses can be found by direct ealeulatio:.

Losses cf thrust of optiiuum nozzles both obtained by means of Shortening the initlal nozzles and constructed from results of the solution of the variational problen, are practically identical.

In wort [56] on the basis of the solution of the variational problem the connection between the marnitude of underexpansion (A—p~2) and angle 8, for nozzles of assirned len-th 4s found:

sin = A—Lt ciga,. * +e! (2.23) ilere a3 — Hach anrle, 1 sing, =F: (2.24)

This dependence can te used in the cnlculation of nozzles of encines onerating in a vacuum.

ео е�"а" =”

2.6. А roximate Method of Construction of a Contour of an Optimum Nozzle

ST

Order of Construction

In the fulfillment of a sketch project for the possibility of the approximate calculation of thrust characterisvics of an engine, the calculation of cooling, and also for the estimate of the weight of engine, it is frequently sufficient to know the approximate contour of the nozzte, the dimensions of which insigni- ficantly differ from dimensions of the exact contour.

In this case it is possible to use methods of approximation of the construction, which permit rapidly findine the optimum contour. below there its given one of such methods founded on results of the solution of the variational problem about finding the contour of the optimum nozzle of assigned lenth and further analysis of the family of found contours [56].

Calculations were conducted with a constant index of the isentrope ng= 1.23. In the construction of the rid of the Characteristics for finding the contour, the region of the critica] section was constructed by arcs of a circle (see Fiz. 2.14). The radius of curvature of entry part on the section prior to the critical section Кем = 0.750; the radius of curvature of the ceneratrix at the entrance into the supercritical part Ген = 0,2250». ТВе зигГасе of the critical Speed was taken to be convex (see Fig. 2.9c).

On the basis of the analysis of the family of contours of optimum nozzles are curves of the dependence of angle в, of inclination cf the contour at the point of contact to the generatrix AA, at the entrance into the Supercritical part and angle of inclination of the contour on section 83 on the dimensionless length of the nozzle L.=L/Re and dimensionless radius of the nozzle on section RyRy» (Fic. 2.13). The Supercritical part of the contour on section А.С 13 approximately approximated by a parabola.

54

Calculations showed that replacement of the exact contour Бу а parabolic contour leads to a diversenes in radial dimensions of the nozzle within 1-32 [56].

a ’ . ie | LT ) , asi 4 5 ° 7 �"" a ; 4 oa | : --- 4 ‘ ||

ее ЗА в

Pir. 2.13. Dependence g and 8. on the

lencth tJM, 4nd rauius of the nozzle on Section ARs:

The construction of a contour is produced in the follewinr oruer (Fir. 2.14). Accordins: to the known Ry te construct a contour of the nozzle in tiie rezion of the critical section, Assuming Rep=lSRe» anc Tem™0,45Rey Accoriins to the known (by assirened or defined calculation) two parancters of the four (Le/Rup: RslRup; вв: бы) with the help of the crid of Fic. 2.13 two other parameters arc

determined. Usually an A (or witeh is the зале, рур;°, ог 9

13) and the angie of inclination of the contour on section Bo. Typicai values of 8, for nozzles of a 7hRD are within 10-14°, Za Sometimes 8, can he cdeternines by the relation (7.23). 9

Бу knowinse Le, Re be ari B3 the fing point Aye drawiny at angle g, a tancent to arc AAs, and point C = accordine to the known

L, and R3- To construct a parabola we conduct fren point C at angle Bb, straight line cr up to the crossing with tanrent Аг. Dividing segments Ant and Cf into 5-7 parts and connecting the corresponding points 1, 2, 3, ete. by straight lines, we construct an enveloping parabola Ac. Line AAC will be the unknown contour of the supersonic part of the nozzle.

Fig. 2.14, Construction of the ecntour of the nozzle.

In the work [8] of Vv. р. Kurpatenkov there is fiven in detail а discussion of the method of constructing the contour of the nozzle,

Examples of Constructinz a Contour of the Profiled jlozzle

Example 1. Ping the profile of a high-altitude nozzle of an engine operating in a уасцит (=).

Pressure in the chamber Ро ® 30 ket /en* (2.96 МИНИ? ); агеа

of the critical section = № сп?; (Вывод 1M); agwis Rew kef-m-der (360 J/kg.deg),; Ty = 3650°%%,

Solution. Let us Set the pressure on the section P3 = 0.075 ket/em® (7350 N/m2),

1. Let us determine the area of the section of the nozzle Г. and also speed У and density of products of combustion on thy section 03. According to formula (1.20) or the rraph of Fig. 1.8,

56

Дом ЛУ о 300 27) д - 28 25; А/А и, > 5.06.

№ (5; =, (=|.

Узы ee ee es.

Accorauing to formula (1.23)

mV eye ]-V stiles) =) om

Let us uefine Po anil p,: Ро

oa т ели вым

ыы лин Кб. о2 < (0.0151 ке кал),

2. Let us deternine tie angle of tnelination of the contour on the nozzle secticn bs which ;roviues the assirned Ps and p the ortinum contour of tie nozzle. Accordine to formula (2.23)

oa BOP Vj OO TT = 0.00, +e! Ft 10-*.31708

whence saaray; fair.

3. Knowing bs and Г ami using mrarhs of Fir. 2.13, let us determine angle 4, “and the lencth of the nozzle L,?

Pa 9PIT; 1-12 4Rsp = O00 a0.

ae:

4, Let us determine the radii of curvature of the entry part of the nozzle:

Reap™ 1EReg@15 0849045 ид; Feng @OMERag 22,6 ди,

5. According $0 вы № and dimensions, let us determine the profile of the nozzle, approximating by the parabola the contour of the nozzle (see Fig. 2.15).

Example 2. Letermine the profile of the chamber nozzle of an engine operating with the following paraneters.

The pressure in the combustion chamber ро = 60 kef/om* (5.92 Mi/m*) ; pressure on section рз = 0.6 kef/em® (0.059 MI") ; area of critical section tay=100 em?) Rup O64 MM; Aes~ 12: Res? Kefem/ke deg (363 J/kg-deg); T, = 3650°K.

solution 1. Since the ambient pressure pis not assicned, we assume that the optimum contour of the nozzle providing the least losses of thrust to friction and dispersion will be at the

VOLUC Gpecn0o, We determine the value of B35 which corresponds to

рес = ее ом,

whence 83 & 10° (15 would be possible simply to assign angle Bs = 10°),

2. According to formula (1.20) or the graph of Fig. 1.8, let us determine fifi»:

= 11,06; Ry/Rey = 3,44; Ry me 194 aes,

3. From the known RyRy and 83 let us determine with the help of graphs of Fig. 2.13 6. and Ly:

4, We construct the contour of the nozzle (Fig. 2.15), .

58

59

2.7. Operation of the Nozzle in Off-Desigfn onditions with High

ounterpressures

It is known that with an excess of ambient pressure Pu over

the design pressure on section P3 the nozzle operates in conditions of overexpansion.

In practice gases are frequently possible when the rocket engine must operate considerable counterpressures of the environment 9-10 and more times exceeding the design pressure on section p,, as, for example, with the underwater launching of a rocket or with the operation near lana of an engine designed for hiph-altitude operation. It was determined that with the operation of a ZhRD under conditions of high counterpressures, there occurs flow separation from walls of the nozzle (Fig. 2.16) and after the Separation point a complicated tystem of shock waves appears. The intensity of the shock and place of flow Separation to a considerable

degree depend on the development of the boundary layer at the wall of the nozzle.

ration from wall at high counterpressures,

The typical pattern of the distribution of pressures alongs the nozzle with flow Separation is shown on Fig. 2.17.

To determine the thrust of the ZhRD, it is important to know

the section in which flow Separation occurs in nozzle (1.e., the pressure at which separation occurs — Por ).

60

Fig. 2.17. Change in pressure in the nozzle with flow зера- ration: 1 = reference plane; 2 — section of separation; 3 — nozzle section; 4 — oblique Shock wave; 5 — pressure dia- gran; 6 — chance in pressures with continuous flow.

Figure 2.10 cives experine.tal curves of the chance in Pressure paltry of the air flow depending upon quantity p/p. and the prefile form of tne nozzle (45).

1 Fic. 2.18. Dependence of рр On pf, ancl the form of the nozzle | with air flow: 1, 2 = profiled

? 1627167 в, <; 3 -- conical nozzle

7 15°. к от

As can be seen from the ziven crap Aim in noz es having a

< fr

hs, «ith the same ratio snaller angle of inclination of the contour 8, flow sepa..tion occurs 2% Jaree values of PalPerp, 12°

»., with a decrease in anile 8. creat overexpansion of the flow in the nozzle is possible.

Qualitatively anzlorous dependences take Place with flow through the nozszle of products of co:

abustion of a liquid fuel rocket engine.

61

о ва ЗБ ЕЕ МЫ

SS Se

Figure 2.19 gives a curve of the dependence of the characteristic 422514255

of flow Separation with expansion On Pa*/p, obtained аз & result of an analysis of the outflow of products of combustion of different

fuels of a ZhRD through a conical nozzle with anple 283 = 30°,

> n © jar | 1,40 Fie. 2.19, Dependence Qos & | 30134 Гиз] 428 on the ratio of nressure : © [Me 0, 1,24 ° “ и ines 423 рр ‹ + щи +В 1429 $ 0 Pu

The characteristic of flow Separation:

rt nN ua —

Knowing Wor, the pressure of Separation can be found:

A. 1-м. (2.26)

For profiled nozzles havine а Maller angle than 8

3» Wantity рырр,, apparently, Will be somewhat snaller,

In the absence of special data expression (2.26) can be used for the determination Of Values per. in the nozzle of a ZhRp.!

STE on caiman Investigations showed that considerable divergences between

experinental and calculation data, obtained DY в. frequently

take place. Tnerefore, One should approach the use of er with great caution,

62

With the known value of Por from elementary considerations we obtain the following formula of thrust of ZnRD:

Риу= «АР.» 7), (2.27)

where P,— thrust created by the part of the nozzle of the engine ur to section a-a and determined by the usual thrust formula.

Pam Ot Soup Bing rd (2.25)

where f,,, — urea of section a-a; 4PUsn»—<) — force actinr on the

Scction of wall of the nozzle; it characterizes the difference

in forces of external and internal pressure on the section of the nozzle a-3., Obviously, the uirection of action of this force is

Opposite to the dlrection of the tractive force.

here

4p=%(p, — 2,,,). (2.29) Coefficient ‘о allows for the degree of restoration of pressure

On the internal wall after the senaration point. It should be estinated dependinc upon the lensth of section 1-3. It Js possible to consider approximately 60 = 0.1-5.5. With an increase in the length of the wali of the nozzle after the Separation point, the

effect of the difference between the external and internal pressure

in the environment of the point of separation is affected less and $0 310414 be taken nearer to the lower linit.

In conclusion let us note that in determining the thrust of an engine in conditions of flow separation, the main question is the finding of the flow separation point a. The position of it

depends on many factors, in the first place, on flow rate ( profile of the nozzle, fuel,

Although there has been condu

M number), and state of the Surface of the nezcle.

eted a larre quantity of experimental

63

eee Заианыт

works by definition of the point of Separation a strict analytic solution to the problem under assumptions acceptable under conditions of work of a ZhRD has not as yet been found.

2.6. Work and Characveristic of Nozzles : with a Central Body

In section 2.1 the definition and basic dlarrams of nozzles with a central body are siven. Let us consider peculiarities of the operation of such nozzles.

Ring nozzles with a central body (see Pig. ?.ld). The throat. of the nozzle has the form of a rinc so that the cas flow is expanded in the annular channel forned by the contour of the central body and external contour. In rins nozzles just as in standard nozzles, tne formation of tne flow is determined by the contour of walls of the nozzle. Losses of thrust to dispersion in the annular nozzle formed by the conleal internal and external contours can be determined by conductinz calculations analorous to those examined above in section 2.2. The pinal expression of @pec for the ring nozzle has the fora

+ (sine, — sinsp «= (бе + Умаве + соке — сия} › (2.39)

where а, - angle of inclination of the internal contour of the

nozzle, 8 — angle of inclination of the external contour of the

nozzle. Obviously, when } Qa=0 the rinz nozzle will be turned

into a round one, and from formula (2.30) we will obtain the

formula of losses in a round nozzle (2.11). In article [8] р the work of ring nozzles with to ancle points is Investizated,

and it 1s shown that rinc nozzles provide rain in the lenmth of the ‘ nozzle.

Characteristics of the operation of rine nozzle essentlally do not differ fron characteristics of a round nozzle, and therefore,

subsequently, in speaking of nozzles with a central body we will

imply only nozzles with a central body having a free limit of

flow.

In contrast to the standar’ supersonic nozzles, in which the low is formed by walls of the nozzle, in nozzles with a central

body the expansion of flow is not limited completely by selitl wall

In nozzles she diarram of which is shown on Fig. 2.le and f the external boundary of the stream is free surface. In plate nozzles (see Fir. 2.16) the free surface is the internal boundary

Let us examine the operation of a nozzle with a central bocy

^

with a free external voundary of the stream (see Fir. 2.le). A

typical siadow photograph of a nozzle with a central body oneratin-

in desimn conditions is snown on 2105. 2.20a, and with an exransion ratio correspondin; to concitions of overexpansion (in the concept

;

accepted for standard nozzles), 1.@., when pa*/Pa<(P2*/Pu)peess — On

rig. 2.20. звазом } flow in a nozzle wit!. cen 3 lesirn operatin.; conuitions; b) ratlo w/e, 1S less than the desirn.

| The expansion and acceleration of flow in the operation of & nozzle with a free external boundary of the stream occurs tn the following way (Fig. 2.21a).

Fig. 2.21. Expansion of flow in a nowzle with a central bouy: a) nozzle with external expansion, b) plate nozzle, } - surface of critical speed; 2? — angle of rotation of the flow w; 3 = free surface of the flow; 4 — rarefaction wave; 5 — contour of the central body; 6 — plate central body; 7 — flow line.

The flow of products of combustion accelerates in the convergent ring nozzle up to a critical speed, which 41s reached da the annular section OA. For a nozzle with full external expansion, further expansion of the fas occurs with the flowing around of the extreme edge at point A where there appears a fan of rarefaction waves and in passing through which the flow accelerates up to speed wand turns at angle w. Quantities м and w are determined by the

pressure drop ps*/p,,. and for the plane flow by formulas (2.37) and (2.38).

With the operation of the nozzle in desi¢n conditions (Ps™ pn) the external boundary of stream AC' 41s directed in parallel to the axis of the nozzle. The arca of the outlet section fy is equal

66

to the area of a circle with radius А; = Вл. With an increase in

the pressure ratio up to (рае) "> (ри /Рь) выч the nozzle operates in conditions of underexpansion. Here due to the increase in А’

the angle of rotation of the flow w» will be increased, and the external boundary of the flow will pass along AC" (Fig. 2.22).

A change in pressure along the contour of the central body will

occur just as in a standard round nozzle (4.e., in the case of a decrease in pg it will remain constant, and in the case of an increase in p§ it will increase proportionally).

Fig. 2.22. Change in free surface of expansion and pressure along contour "Omc" under different operating conditions of nozzle.

The equation of thrust of a nozzle with a central body when operating in conditions of underexpansion will have the same form as that for a round nozzle:

р =. 0+ в.) (2.31)

The characteristic of the change in thrust of a nozzle with a

central body when operating in conditions of underexpansion will

coincide with the characteristic obtained for a standard nozzle (Fig. 2.23).

67

ee

Fig. 2.23. Characteristic of the change in thrust for different ; types of nozzles: 1 — ideally . variable-area nozzle; 2 — nozzle with a central body; 3 — round Orerexpecsicn | Unde reaparsion nozzle with flow separation;

4 = round nozzle without flow separation; 5 — design conditions.

With values of p%p, smaller than the design values, the flow in the nozzle with a central body is different from the flow in а standard nozzle: expansion on the edge at point A occurs only up to the ambient pressure рь The extreme rarefaction wave fron point A enters into the contour 0C at point m, and further expansion (overexpansion) up to pressures less than fm» does not occur so that the external boundary of the flow formed by the external pressure will pass along the line AC''' (this is well evident on the Shadow photograph of Fig. 2.20b). The expansion of flow alonc contour OC up to section m-m occurs just as in the standari nogzle, and the pressure at point : can be considered equal to the ambient pressure »,. With the flow below point n flons the contour mC flow turns ond is somewhat compressec (as with the flowing around of a concave surface by supersonic flow). At point m (see Fig. 2.22) the rarefaction wave Am 1s reflected fron the wall. The reflected shock wave mii at point :! 13 arain reflected from the external free boundary on the surface, ard the rarefaction wave, entering into point R of the contour of the central bory will again be reflected and so on. With раззаге throuch the shock wave mN the pressure is somewhat increased, in consequence of which on the section of the wall mR the nressure on the wall in reneral can even increase somewhat. ‘Then with passace throurph the rare- faction wave NR the pressure arain drops, and up to the line of the end of the nozzle CC" the flow will pass consecutively through & system of several waves of rarefaction and compression. Therefore, below point m the pressure along contour mC can be considered constant and equel to the ambient pressure, as is stown on Fir, 2.22. Thus in conditions corresponding to conditions of overexpansion

68

for the standard nozzle with operation of a nozzle with a central body, overexpansion does not occur. The flow will be expanded only up to a pressure equal to the ambient pressure. The area of the outlet section of the flow with operation with (ps°/ps)<��<(ps*]ps) secx will decrease and will be equal to the area of flow in section mem.

The thrust of an engine having a nozzle with a central hody | operation at a pressure rat1o (ра°/р)” < (рз*/рь) вк, Will correspond to design conditions of an engine operating with expansion ratio (mlm) 1.е.,

Pra Sh. Г % (2.32)

Where ю» — speed corresponding to the civen ratio (ps%/p,.)”.

thus, with a decrease in the pressure ratio ~:°/p, lower than the design ratio, the thrust of an engine with a nozzle with a central vody will correspond to the thrust of an engine with a round ideally variable-area nozzle. Therefore, nozzles with a central body with a free surface of expansion are frequently

called self-adjusting: nozzles.

The characteristic of the change in thrust of the nozzle with a central body with the operation with (pa*/Pa)�<’< (P2°/Pa) pace in accordance | with equation (2.32) will pass higher than the characteristic of the standard nozzle, in which because of the overexpansion of flow tne thrust will decrease sharply.

It is necessary to note that under off-desifn conditions of tne operation of the engine with a nozzle with a central body, besides a change in speed (because of the chance in ра“/рь) there also occurs a deviation of the directions of motion of the flow from the axial. This leads to certain losses in thrust to dispersion, in consequence of which the characteristic of the nozzle with a central body will pass below the characteristic of a round ideally varlable-area nozzle.

seb ty же

| в

69

Nozzle with Partial Internal Expansion

Figure 2.ld gives a diagram of a nozzle with a central body with partial interne! expansion. Such nozzles are an intermedtate type between the ring, nozzle and a nozzle with a central body and external expansion. The application of ‘hem is expedient in those cases when the nozzle with a central body should provide acceleration of flow up to high M numbers, i.e., with a large ratio p/p, he fact is that in a nozzle with full external expansion with an increase in p/p, angle of rotation of the flow w is increased. At values of pa’/P corresponding to values of M of the order of 3.6-4.2 (depending upon the index m,), the angle of rotation of the flow reaches 90° and grows with a further increase in ра“/ра (Fir. 2.24).

Since 1t is necessary to provide a sufficiently sooth acceleration of products of combustion up to the speed of sound, it is obvious that with an increase in the angle of rotation of the flow there will be an increase in the diameter of the annular combustion chamber, i.e., dimensions and weight of the entire engine. ‘Therefore, at pr*/ma for a decrease in the anele of rotation of the flow (and consequently, dimensions of the enrine), it is expedient to use the preliminary expansion in the ring nozzle up to a certain speed Mj (1<Ms<M).

Ее

Мо Ме Му

Pir. 2.24. Charge in the angle of rotation of the flow and position of the surface of critical speed OA depending upon the tl number.

Let us consider how the characteristic of the change in thrust for a nozzle with partial internal expansion will pass (Fir. 2.25).

70

Fig. 2.25. Characteristic of a nozzle with incomplete internal expansion.

| | 1 | i

м“

For a nozzle with a design relationship of pressures equal бо (ра*/рь) Пре, the characteristic II with full internal expansion (standard round nozzle) will pass along curve ecd and with full external expansion — along gbcd. For a nozzle with full internal expansion designed on (ps*/pu)lpcy Characteristic I will pass alone curve abf. With the application of a nozzle with partial internal ana further external expansion the chanre in thrust with an increase in [2°/Ры до (рз*/ри) = Will pass along characteristic I. However, with a further increase in ps*/p, the nozzle will operate as a nozzle

with external expansion, and the characteristic will pass along the line bed.

Thus, curve abcd will be the characteristic of the chanre in thrust of the nozzle with partial internal expansion.

Plate Nozzle

In a plate nozzle (see rig. 2.lg, 2.21 anc 2.26) the annular critical section of the nozzle is located nearer to the axis of the nozzle, which permits decreasing the dimensions of the combustion chamber. Products of combustion flow out from the throat, parting in a direction from the axis.. With flow along the external contour oi nozzle OC the flow turns in a direction along the axis. Gas expansion occurs with the flowiug around of edge A of the plate central body. The free surface is the internal border of the flow whose contour is determined by pressure ра’ at the end of the plate of the central body. In general this pressure is somewhat less

71

ТЕ

атаке:

=;

eS ый

than the ambient pressure Px in connection with the ejection of flow fron the central region of the nozzle.

Fle. 2.26. Engine with a Plate nozzle (55].

The area of the outlet section of the plate nozzle orerating in design conditions is equal to

Syma (Ri— Ri). (2.32)

Figure 2.27 shows the location of a free internal surface of flow in the plate nozzle for different relationships of Ра * р"�":

a) desicn relationship; b) underexpansion; с) conditions correspondinc to conditions of overexpansion.

When ры’) < (ра �"") ас just as in the case of the nozzle with а central body, on section mC the pressure on the wall of the nozzle

with transition through the system of reflected waves can increase somewhat. и

In contrast to the nozzle with a central body and external free surface in the plate nozzle in fonnection with the fact that Ра’ <рь,

72

a@ certain overexpansion of flow at the wall of the nozzle up to

pressure p,.�<� 1s possible, i.e., in the Plate nozzle a negative component of thrust owing to overexpansion can appear.

Fig. 2.27. Change in free surface of expansion and pressure alone contour OmC under different operating conditions of the plate nozzle.

however, the magnitude of such overexpansion is small, and in the first approxtiaation it 1s possible to assume that in the plate nozzle overexpansion is absent. In this case the characteristic of the change in thrust depending upon pip, for the plate nezzle, will be determined by the sane cquations as for the nozzle with 4 central vouy and will pass as 1s shown on Fig. 2.23.

2.9. Calculation of dozzles with a Central Lody

‘Let us examine the approximate method of the construction of a contour of nozzles with a central body. For simplification of calculation let us assume that the expansion of flow in a nozzle occurs Just as with the Prandtl-Mayer plane flow. Therefore, in the calculation of the expansion of flow after edge A, we will consider the dependences obtained for the plane flow applicable.

73

een < ~ мнит ричи зы ыы зы

Calculation of a Nozzle with a Central Body with External Expansion

Let us consider that we know the fuel, 1.e., am, temperature То and pressure in the chamber Po and also the expansion ratio P3/P8 (or, which is the same, M5). The subsonic part of the nozzle is designed in such & manner that on the edge of the nozzle at point A the critical Speed was sct, 1.е., МА = 1. Then OA is the surface of the critical Speed (Pig. 2.28),

Fig. 2.20. Calculation of the contour of a nozzle with a central body: 1 = axis of the nozzle; 2 - free surface,

Let us determine the position of ede A and the angle of inclination of the edre. With the flowine around of the edre there will be formed a bundle of waves of rarcfaction. With desien conditions, according to conditions of the operation of the nozzle the direction of flow on the boundary of the free surface AC! should coincide with the direction of the axis of the nozzle. For this the ancle 03 of inclination of tangent Ae to the contour at point A should be equal to the anzle of rotation of the flow with the acceleration of flor fron ity = 1 up to the assigned sreed Ч; (1.е. with the expansion of flow up to P= pe):

> ис / ив УМРЕТ. (2.31)

74

The distance of edge A from the axis of the nozzle is equal to the radius of the outlet section Rs and is determined by formula (1.21),

| ei EV [b(t мет, (2.35)

wherc

ee ee

f=aR, (2.36) and fg, is determined by the relation (1.42).

To construct a contour of a central body OC, let us find the position of the arbitrary point b on the contour of the central body. Lntering into point Bb will be the intermediate rarefaction wave AB. The flow rate w (or M) along the rarefaction wave ir (characteristic) AF is constant. The section of the flow f in which the speed is equal to w will be defined as the projection of the lateral surface of a frustum of a cone formed by the rotation of segment Ab around the axis of the nozzle on the plane normal to the speed. The area of the section of flow f and anrle w between the direction of flow a.d tancent Az are determined ky relations

| LEV eh (ety т

"ИЕ efi arctg ЗЕЕ (мет Г 1)—arcig VMF—T, (2.38)

egy apenas yore

Angle a between directions of characteristic AB and speed w (‘ach angle) will be determined by the relation

singe (2.39)

75 ’

С поль ИИ Воть rn лем dal tatiana aes ss ata nntnap titanate к аи би И

If R is the distance from point B to the axis, and lenrth AP is equal to L, then

Smal (R+R,)sinc. (2

Designating ф the angle between the direction of tanrent Ae and normal to the axis, from geometric relationships we can write

9)

= 90° —4; р ZBAC' m(a—e) +0, —0— (04-4) +90". (2.42) Designating ZBAC’mé, we obtain $90 +a—(0+7); . (2.12) СОВА 90° —1 = —в-+(.-+ у). Since BD = В; - R, then —R ~ es fe—e+O (2.43) Substituting expressions (2.43) and (2.39) into equation (2.45), we obtain f= я (4—*) Мова (« +4)" (2.48) whence, taking into account relation (2.36), к 1-£ M cos (a —(w +-«)]. (2.45) Substituting equality (2.45) into formula (2.43), we obtain учет ЕЕ [1 и! ee I~ F нон - + enn)

76

or

L 1 I~ - [-У ie Г Мои (+44 aes

Equations (2.42) and (2.47) are design equations for the construction of the contour of a centrai body.

The position of extreme points of the contour C and 0 can be determined immediately. At point C

ome; MeM,; o=ma,; = ь сов в (сов — (+90 ~0,))msing,, | (2.48)

Substituting (2.48) into (2.42) ana (2.47) we obtain se = о; М (2.489) At point 0

o=0; c=; Mal; far/,,. (2.50)

Let us substitute equality (2.50) into expressions (2.42) and (2.47). Then

6,, = 180° —4;

1 2? > Abe ты} | fs ao 3] (2.51)

L,=OA =I, Ry

To construct a contour of a central body having assipfpned a series of ii values from M #1 to Ms M3 it is convenient to reduce the calculations by definitions of 9 and L to a table (see Table 2.3).

Shortened Nozzles with a Central Body

As a result of the calculation we obtain a contour of a central body of an ideal nozzle, f.e., a nozzle providing on the

77

‚о: дн�& ae

section a flow which is uniform and parallel to the axis of the

nozzle. Thus, just as in the case of round nozzles, an ideal

nozzle with a central body has very great lenrth. Therefore, here

for a decrease in dimensions and weight it is expedient to use

shortened nozzles. Experiments show that the shortening of a central

body by means of replacement of the profiled contour of a cone г leads to very small losses of thrust (55].

A comparison of profiles of the central body of an ideal and shortened nozzles, bounded by conical contours with different г half-angles of opening of the cone, is given on Fig. 2.29a, from which it is clear that replacement of the end part of the contour by a cone with an angle of opening of 20-30° permits decreasing . the length of the nozzle by 40-607.

Fig. 2.29. Shortsning of nozzles with a central body: a) replace- ment of part of the contour by a cone; b) change in ¢, depending

upon the half-anzle 8 of the central body; c) segment of the part of the contour.

|

78

}

Figure 2.29b shows the change in the coefficient of the nozzle $. depending upon the half-angle 6 of opening of the cone. From

the graph it 1s clear that even when в = 30° losses of thrust are ubout 1%.

The central body can be shortened also by cutting its end Fart and replacing contour ABC by contour ABD (Fic. 2.29¢).

\ shortened contour of the nozzle with a central body can also

be constructed by solving the variational problen on the detecting of the optimum contour.

Calculation of Plate Nozzles

Having taken for the Plate nozzle the assumption about the Plane flow of gas with the flowing around of edpe A, the afore-

mentioned method can be used for an approxinate calculation of the Plate nozzle (Fig. 2.30).

Fig. 2.30. Calculation of the contour of a plate nozzle: 1 — axis of the nozzle; 2 = free surface.

Initial data for the calculation of the plate nozzle are the Same as in the calculation of a nozzle with external expansion.

79

Let us designate: R, - radius of the plate central body fron the axis to edge A (this radius is assigned from design considerations); R, - padius of the contour of the nozzle section. Then

Soma (RE— Ri). (2.52)

For the arbitrary point B the contour of the nozzle arena of the section of flow with speed w (or 1) will be defined as the projection of the lateral surface of the frustum of a cone formed by the rotation of segment AP #= L around the axis on the surface norzal

to the direction of the flow rate w:

Soal(R+R,)sine. (2.

uv’ ay ~~

Since in this case relation (2.42) are real, then

tea (2.59)

Substituting equalities (2.54%) and (2.42) into expressic: (2.53), we will obtain

(ею)

ты 6+9 (2.55)

vnence

RoR +в 6+9 (2.56)

Substituting equality (2.56) into expression (2.54), we will obtain:

be Scan (V+ romeo + - 1); (2.57) Z я ети + oe coe fa— 6-9 -|. (2.58)

80

Example of the Calculation of a Nozzle with a Central Body

Determine the basic dimensions and contour of the central body of a nozzle with external expansion.

The pressure in the combustion chamber p§ = 60 kef/em* (5.92

>

Lif"); the design ambient pressure p,=pade КЕР/ст? (0.048 МНИт? ); the temperature in the combustion chamber �>> = 3050°9К; вне 1пдех ог the isentrope й:=12; the fas constant of products of combustion

В = 37 kef+m/kc-deg (363 J/kg-deg). ‘The theoretical thrust of the encine P = 10 7 (9.52-10' ий). Losses are not considered,

Solution 1. Бу formula (1.23) we find the Ny number of the ee a flow:

И | -

2. we find ЛИЛ:

14 Sm м om 141-31 р ats 1 1,2+1 Jan № mt 3.5 =

3. The specific thrust Py, and fuel consumption 0: e-ty/ a | а Е ]- Pow ве м"�", р (я)

wi V s0n иж [1-2 Jam Бато (3940 воЛа),

On Os kg/e.

81

‘mI DS ene с

Rs nt es ie Aan aad tan

Ц. The complex 6 and area of critical section fy.

ыы

и = 112 cat,

5. The area of section 23 and radius В. : fam 14,6fcy = 1643 ва? „-/Ё -Укм-= soa

6. The angle of inclination of the external edre to the axis of the nozzle Wy and angle y:

“= И acu чу (и!) ив УЖ

isi Pe ey) een Г 8.30 — ме 3.9—1�< 77°26°; ф = 00° — ва = 12°34°.

7. Let us determine the contour of central body. For this by formulas (2.42) and (2.47) we find © and L, woich correspond to various values of i1. Data of the calculation are riven in Table 2.3. Here values w and jfi,, are determined or accordine to the available tables or graphs or by formulas (2.37) апа (2.32). A constructed contour of the centrai body of an ideal nozzle is shown on Fig. 2.31, from which it is clear that its lencth was obtained long.

bop

¥ ‘Pree surface

Fig. 2.31. Example of the construction of a con- tour of a central body.

82

For the shortening of the central body we replace part of the profiled contour by a cone with a half-angle of openinr of the cone 6 = 30°. Then the renerarix of the shortened central body will be line OBD, and the length of the shortened nozzle is 55% less than the lenrth of the ideal nozzle of a central hody.

Advantares and Deficiencies of Nozzles with a Central Pody

Investigations of nozzles with a central body have herun relatively recently. Therefore, it is possible to conduct only a preliminary analysis of their basic possible advantares and deficiencies.

The basic merits of engines with a nozzle with a central роду are tne best characteristics of the change in thrust and smaller dimensions than for encines with a round nozzle. Аз was noted above, the reason for this is their ability of self-rerulation in Operating conditions corresponding; to conditions of overexpansion in round nozzles.

The length of the enrine with a shortened nozzle with a central hody is considerably less than the length of the standard епг1пс. Furthermore, the cavity of the central body can be used for the Gistribution in it of units of installation (for example, turbopunr unit [TNWA]) which also permits decreasing the dirensions. Firure 2.32 shows a model diagram of the distribution of units in the Cavity of a central body, and a comparison of a ZhRD with a round nozzle and with a nozzle with a central body is given. The expected advantage of nozzles with a central body is the possibility of controlling the thrust vector by means of dividing the annular chamber intc separate Scgments and changing the fuel consumption in each of the separate chambers. Incidentally, with this it 1s possible to expect that due to the dividing of annular chamber into chambers of smaller dimension, the probability of the appearance of vibration burning also will decrease.

83

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ПРА АТА И ПИ Ч АНИ _

84

Fig. 2.32. ‘iodel diagram of the distribution of units in the cavity of a central body (shown on the left for comparison 1s a single-chanber engine of the same thrust): 1 = Turbopump unit; 2 — auxiliary tanks.

Of the deficiencies of nozzles with a central body, in the first place it is necessary to note the difficulty of organization of the cooling of engines with such a nozzle because of hirhly developed perimeter of the thrort and complexity of feed of the coolant (see section 4,13). Difficulties of cooling sonewhet decrease with the use of a plate nozzle.

In comparing plate nozzles with nozzles with external expansion, it is necessary to note as an advantage of plate nozzles the lesser effect of interferences from the side of the environment, which appear with moverent of the rocket in the atmosphere. The weight of the engine with a plate nozzle оп the whole somewhat decreases because of the considerable decrease in weight of the combustion chamber. However, @s we noted above, because of the ejecting action of the flow in plate nozzle. overexpansion up to

Pu�<� 18 possible, which leads to an impairment of thrust characteristics.

85

i

CHAPTER III CARBURETION AND CHAMBERS HEAD OF A ZhRD

The efficiency of the combustion chamber of a liquid fuel rocket engine (ZhRD] to a considerable degree depends on organizing the processes of carburetion and combustion, which in turn essentially depend on properties of the fuel and construction of the chamber head of the engine.

3.1. Basic Elements of Carburetion and ombustion Processes

Carburetion and Conversion of Fuel

Garburetion is defined as the complex of processes taking place from the moment of introduction of fuel components into the chamber

to the formation of a uniform mixture, Thus, the process of carburetion is composed of the supplying of components to the chamber through injectors, breaking the streams into drops, evaporation, and mixing.

Process of carburetion should ensure:

1. Maximum combustion-of components in the chamber,

2. The most homogeneous distribution of the proportion of

components (1i.e., a) and flow intensity r per chamber section (see section 5,1).

3. Minimum tendency to appearance of unstable burning,

86

ba a

4, Formation of a boundary layer necessary for shielding the walls of the chamber from high heat flow into the wall.

Conversion of fuel into products of combustion will be defined

as the total process of the combined processes of carburetion and combustion of fuel,

Processes of ccnversion depending upon tape of fuel (hypergolic or nonhypergolic) are schematically represented in Fig. 3.1.

Fig. 3.1. Diagram of flow of conversion processes,

With hypergolic fuel, after atomization of the components, the process of combustion proceeds in three directions. Part of the fuel is evaporated prior to mixing in the liquid phase, After that mixing of components occurs, in the gas phase and their combustion, which leads to formation of the products of combustion,

87

REA Ry

The second part of fuel 15 mixed in the liquid phase, In the contact between drops of fuel and the oxidizer the burning reaction Starts directly in the liquid phase. In connection with the sharp increase of temperature during burning, part of the fuel, insufficiently mixed in the liquid phase, is evaporated without Succeeding in entering

into the reaction. Further mixing and combustion of this fuel occurs even in the gas phase.

Simultaneously with homogeneous, there can occur heterogeneous burning, 1.е., burning of liquid drops of one of the components in gaseous vapors of the other, Such burning can take place in case of advance evaporation of one of the components or during formation of

very large drops as a result of atomization which evaporate more Slowly than small drops,

The burning reaction in the liquid phase is characteristic only for hypergolic components,

With nonhypergolic components, the reaction in the liquid phase does not appear, and the conversion process proceeds according to the diagram in Fig. 3.14b. After atomization of components their mixing occurs in the liquid phase, In so doing, preliminary mixing is possible (for example, in an air-boost system), Then there occurs evaporation of fuel components, their further mixing and reaction,

Simultaneously there also takes place heterogeneous combustion of drops,

According to the flow of the processes determining the conversion process the combustion chamber of a liquid fuel rocket engine can be divided lengthwise into the following basic zones (Fig. 3.2):

I — atomization zone, ee a

II — evaporation zone. III — mixing and combustion zone,

88

Fig. 3.2. Diagram of flow of processes in the combustion chamber: 4 — atomization; 2 — evaporation;

— mixing and burni } — К1пе% {с burning; 5 ~ diffusion burning; 6 —

incompleteness of combustion,

It must be noted that division of the chamber into the zones mentioned is somewhat conventional since the processes of atomization, evaporation, mixing and combustion, do not GO on in strict sequence, and it is impossible to Say that evaporation starts only after full atomization, and after evaporation — mixing, etc, In each zone partially simultaneously two or three process take place. Thus, the name of the zone determines only the process most characteristic for it. Let us sort out the flow of processes in each of the zones,

Atomization

The basic indices characterizing the quality of atomization are fineness and homogeneity of atomization, and also the range of the stream and distribution of flow intensity along its section,

The quality of atomization of the fuel proceeding into the chamber determines the flow of further processes in the chamber — evaporation, mixing and Combustion of fuel. ‘The processes both in the fuelpassing system and injectors, and also in combustion chamber have an effect on the quality of atomization, The type of injectors (jet or centrifugs1), the construction and their mutual location to a considerable degree predetermine the process of atomization of the fuel,

89

Disintegration of the stream into drops and splitting of drops occurs after exit of the stream from the injectors under the action of external and internal forces,

External forces (sometimes called serodynamic) include forces of interaction of the Component with the medium into which it is injected. The magnitude of these forces depends on environment, velocity of the Stream, and dimensions of the drops of liquid, External forces also include the force of interaction during crossing of streams or during encounter of the Stream with the hard wall, With an increase of velocity of the stream with respect to the medium in which injection occurs, the action of external forces grows, whereby disturbance of the surface of the stream grows, which leads to the most rapid splitting of the Stream, and consequently also to improvement in the quality of atomization,

However, disintegration and Splitting of the stream can occur in the absence of external forces, Thus, for example, during injection

of a component in a vacuum, atomization occurs under the action of internal forces,

Internal forces include the action of turbulence and molecular forces. In a stream of a component emanating from the injector, turbulent pulsations or velocity appear; inside the Stream, incles of liquid accomplish random motion, Intensity of the turbulence depends on pressure drop at the injector, density, viscosity, and also on the design of the injector, Increase of pressure drop (i.e,, increase of exit velocity) promotes an increase in intensity of turbulence, in consequence of which Splitting of the stream is accelerated, 1.e., the quality of atomization is improved, Molecular forces — these are forces of viscosity and surface tension,

During exit of the stream from the injector, external forces and turbulent pulsations of velocity inside the stream strive to break up the strean (liquid sheet) on the other hand, the force of surface tension and force of viscosity prevent splitting, Thus, splitting of the strean and formation of drops occurs in the following manner,

90

During the injection of the component through the injector there will be formed a liquid stream (liquid sheet). Under the action of external forces and turbulent pulsations in the stream disturbance appears on the surface of the liquid. As a result of further action

of external and internal forces on the component there will be

a film of liquid and the stream (liquid sheet) breaks up into

formed

particles of different size and shape. Small particles under the action of surface tension take the shape of a sphere and will form drops; the large ones continue to disintegrate further (Fig. 3.3).

Fig. 3.3. Mechanism of disintegration of stream (liquid в a) jet injector; b) swirler; c, d) distribution diagrams of flow intensity, 1 — 8tream; 2 — disturbance of surface and formation of connecting necks; 3~ splitting of streams

(liquid sheets) into parts; 4 — liquid sheet,

Fineness of atomization is characterized by the size of the average

diameter of the forming drops.

Homogeneity of atomization is characterized by the distribution

curve of drops of a given diameter,

91

|

Fineness and homogeneity of atomization depend on properties of the component, design of the injector, and alsc on the parameters of operation of the combustion chamber in which injection is produced, and in the first place on the density of the products of combustion, With an increase in density of the medium, in which injection is produced and also with an increase of pressure drop at the injector, fineness of atomization is improved, For a liquid fuel rocket engine, the average diameter of a drop is within limits of 25-250 microns,

It is interesting to note that for a diesel engine having considerably Greater pressure drop at the injector, the average diameter of a

drop is less and is within limits of 10-100 microns, Basic cause

of this consists in the fact that in diesel engines injectors are set with considerably greater pressure drop APy than in a liquid fuel rocket engine,

The range of the stream (or depth of penetration) is determined

by the exit velocity of the stream from the injector (1.e., pressure drop at the injector), direction and angle of atomization, and also density of medium in which atomization of components occurs, An increase of range is undesirable since it increases the dimensions of the atomization zone, and consequently also the dimensions of the combustion chamber,

Distribution of flow intensity of component over the cross section of stream is determined primarily by the type of injector,

Typical distribution diagrams of flow intensity for a jet and a swiricr are shown in Fig. 3.3c, d. For a jet injector the peak of flow intensity at the axis of the injector is typical, For section of stream of a swirler the presence of two peaks a certain distance from axis is characteristic, .

As the stream is removed from the nozzle of injector a certain smoothing of peaks occurs (dotted line on Fig. 3.3c). Usually it és assumed that flow intensity along the circumference of stream is constant, aithough in reality flow intensity is nonuniform,

р Ч

Evaporation

The process of evaporation is an important stage in the preparation of fuel for ignition and combustion, since the basic quantity of fuel in the chamber of a ZhRD ignites and burns in the gas phase, On the reaction rate of the evaporation process strongly depends the total time elapsing to formation of the products of combustion. The complexity of the evaporation process in the chamber of a ZhRD is increased by the fact that here simultaneous evaporation of not one component occurs, but a complicated mixture of components with different physical chemistry properties, Thus the evaporation process shoul¢ be completed after a very short interval of time on the order of 0.002-0.008 s [66]. The heat necessary for evaporation of drops in the chamber of ZnhRD is supplied from the combustion zone primarily by means of convection heat radiation fron reverse currents of hot gases, Reverse currents appear as a result of ejecting action of the stream of injected fuel. The stream of the component attracts to itself gases found in cavities between the streams, Into the zones of rarefaction thus formed new portions of hot products of combustion proceed (Fig. 3.4).

—— Reverse currents

Fig. 3.4. Appearance of reverse currents,

Part of the heat is supplied to the adrops by radiation from the core of the flame, Furthermore, the supply of heat occurs also by means of the combustion beginning in the evaporation zone,

The rate of heating and evaporation of drops, and consequently also the length of the evaporation zone depend on ambient temperature, dimension of the drops, relative velocity (to flow of gases) of movement of drops, and also on physical chemistry properties of the substance of the drop.

An increase in temperature of the medium accelerates the evaporation process; an increase of pressure, conversely, somewhat retards it. А decrease in dimension of a drop and increase in relative velocity of a drop leads to acceleration of evaporation,

For an illustration of the influence of the dimension of relative velocity and properties of a liquid let us examine the graph (Fig. 3.5, 3.6, 3.7). In Fig. 3.5 there are graphs of change along the length of the combustion chamber of temperature of drops of heptane of different dimension. We see that, with an increase in dimension of drops, the length of section of heating of a drop to boiling point grows. Thus, for a decrease of the evaporation zone it is desirable to have finer atomization,

rs Fig. 3.5. Change in temperature of heptane drops of different dimen- sion along the length of the combustion chamber, ve $ 49 9 це

In Fig. 3.6 there is shown the change in velocity of drop We and gas Wp along the length of the combustion chamber.

Velocity of drop Мк, in the beginning is delayed since there occurs deceleration of the drop in a gas medium having slower velocity “г. As the rate of evaporation of the drop Wp Grows, and with Ly > 25 wa exceeds the velocity of the drop. Now already the flow of gas

94

attracts the drop to itself, which leads to an increase in its

velocity. Thus, the relative velocity of drop (Wy - Wy) initially

Grows. However, as the drop evaporates, in connection with the decrease in dimension of the drop and decrease due to this influence of forces

of inertia on velocity Wy» relative velocity (wp - Wy ) again decreases, The quantity of evaporating liquid Syon &t first grows slowly, but

in a region of high relative velocities (Ww, = Wy) there occurs strong growth of Висп*

ok

-8 к = -

"|

— a

_-

9

|

9 аи м Я По Fig. 3.6. Change of parameters of evaporated drop of heptane and veloc- ity of gas along the length of the chamber,

The effect of physical properties on change of temperature of drop Ty 18 shown in Fig. 3.7a and on rate of evaporation — in Fig, 2.76. We see that drops of low-boiling components 05 and Fo attain boiling point considerably quicker than drops of other components,

In accordance with this, for evaporation of low-boiling components

a smaller section of length of combustion Chamber is necessary, Comparison of graphs 1 and 3, Fig. 3.7b shows also that rate of evaporation depends not only on boiling point, but also on heat of phase transition of liquid into vapor. At an assigned pressure of ро = 21 Ка /ст? ("га Mn?) the boiling point of heptane (curve 1) is higher than the boiling point of ammonia (curve 3). However, the heat of evaporation of heptane (52 kcal/kg) (217 kJ/kg) is considerably lower than the heat of evaporation of ammonia (29 kcal/kg) (1125 kJ/xg), and due to this, full evaporation of heptane occurs more rapidly, in spite of the higher boiling point.

95

face %

Cw www | AS Wises s ww es я a) Е om ь) by cm

Fig. 3.7. Change of T, and исп of drops of different components: 1 — heptune CoHag 3

2 — hydrazine NoHy 3 3 — ammonia NH. ; 4 — oxygen 05; 5 ~ fluorine Бо.

We examined the effect of different factors on evaporation of a separate drop. In the process of eveporation in the combustion chamber of a ZhRD, as was indicated, there takes place simultaneous evaporation of a multicomponent mixture of drops of various dimensions. Thus the process of evaporation is essentially complicated as compared to the process of evaporation of individual drops. Irregular- ity of atomization leads at the beginning of evaporation to formation of a large quantity of vapors at the expense of rapid evaporation of fine drops. However, further evaporation of the remaining bigger drops is protracted and on the whole the period of evaporation of a mixture of drops of different dimension is greater than the period of evaporation of a drop of average dimension, The greater the irregularity of atomization, the more the period of evaporation of a mixture of drops is increased.

During evaporation of drops of multicomponent liquids which frequently takes place in a ZnRD, according to the evaporation in a drop concentration of the higher-boiling component is increased, which furthermore can lead to protraction of the process of evaporation.

Besides irregularity of atomization and the presence of multi- component drops of liquid, on rate of flow of the evaporation process of fuel during use of hypergolic fuels the presence of reactions in the liquid phase appearing in the zone of evaporation also has an effect. Heat thus realeased can Significantly accelerate the evaporation process,

Mixing and combustion

As a result of atomization and evaporation a vapor of fuel and oxidizer will be formed, from which a fuel mixture is obtained, Mixing of fuel and oxidizer occurs both in the liquid and also in the vapor phase. The most desirable, but also very difficult to achieve in a ZhRD is a method of mixing — formation of a monopropellant or emulsion by means of complete preliminary mixing of componer.ts in the liquid phase just prior to introducing them into the combustion chamber, For a ZnRD, mixing of components in the actual combustion chamber is characteristic. Intensity of mixing of components is determined by turbulent diffusion,

The process of mixing the fuel components in the chamber of a ZnRD begins directly from the moment of their introduction into the combustion chamber and is finished only as the combustion of the fuel. In zones of atomization and evaporation there occurs mixing of part of the components both in the liquid and also in the gas phase. By means of flow of reactions of combustion directly in the liquid phase (with hypergolic components), and also by means of the intense supply of heat addition from the flame front (which also leads to the appearance of first sources of burning of the ¢18 phase still in the zone of evaporation) process of mixing in these zones is considerably intensified, However, due to irregularity of atomization and different rate of evaporation of components, the process of mixing is not completed in these zones, but continues further in the zone of mixing and combustion. Consequently, part of the fuel enters this zone after evaporation and mixing, i.e., in a form completely prepared for combustion of a mixture in which the reaction of combustion already has partially begun. During combustion of this part of fuel, formation of the flame front begins.

я

aA

The remaining fuel (a large part) enters the zone of mixing and combustion basically in an evaporated form, but still insufficiently mixed for combustion (even introduction of separate liquid drops of components is possible), Further mixing of this part of the fuel occurs. Owing to the great difference between the temperatures of the flame and the entering components, the intensity of mixing in this zone 18 very great. Simultaneously in zone of mixing and combustion there occurs combustion of the fuel already mixed. Due to such "parallel" flow of the processes of mixing and combustion, the flame front in the combustion chamber of a ZhRD is not any clearly outlined surface separating the prepared fuel mixture from products of combustion, Width of the flame front is determined by fineness and homogeneity of atomization, volatility of fuel components, and intensity of turbulent diffusion determining, in turn, the rate of flow of the mixing processes,

The process of combustion of the atomized fuel can be divided schematically in two: mixing of components and chemical reaction,

Rate of flow of the reaction is determined dy temperature and activation energy. At low temperatures, chemical reaction rate is relatively low and less than the speed of mixing of components,

Rate of flow of the burning process, on the whole is thus determined by rate of the chemical reactions, Such burning, whereby the speed of the process is limited by the chemical reaction rate, is called

kinetic burning.

In the combustion chamber of a ZhRD kinetic burning can take place only in the very beginning of the zone of mixing and combustion, where the temperature is still comparatively low. Let us limit this region of the zone of mixing and combustion to the section m-m (see Fig. 3.2) and call it the region of Kinetic burning. After section mom high temperature is developed, Here already the chemical reactions flow practically instantly. Thus the rate of the burning process is determined by rate of mixing. Such burning, with which the rate of the process is limited by the rate of the mixing process, is called diffusion burning, Therefore, the region after section mom we will call the region of diffusion burning. In the combustion chamber of a

98

ZnRD, the region of kinetic burning is very small and diffusion burning plays a decisive role, Therefore, it is frequently assumed

that, on the whole, diffusion burning takes place in the chamber of а 218),

In Fig. 3.2 the graph of change of quantity of burning fuel along the length of the combustion chamber does not reach a value of G, /G. = 1. It is obvious that the value 6,/а; determines by itself the degree of physical combustion efficiency. Accordingly difference 1 - (6,/c”) characterizes incompleteness of combustion, For a ZhPD combustion efficiency Pop = 0.95-0.99.

Curve and conversion and time of stay

The degree of completeness of conversion of fuel into products of combustion is customarily estimated by the change of combustion efficiency along the length of the combustion chamber or in the time elapsing from the moment of introduction of components into the chamber, The curve characterizing change of combustion efficiency along the length of the chamber of in time is called the fuel conversion curve (burnout curve). Sometimes the conversion curve is constructed in the form of change of temperature or specific volume of the mixture of components (fuel or products of combustion) along the length of the chamber or in time,

An example of a conversion curve is shown in Fig. 3.8a, where there is given a typical qualitative dependence of change of specific volume along the length of the combustion chamber, Specific volume Vo corresponds to calculated specific volume with full combustion. The conversion curve does not reach Vos which is a result of incomplete- ness of combustion, Within the limits of tolerated incompleteness of combustion point 2 corresponds to completion of process of conversion. In Fig. 3.8b there is given a conversion curve showing change of specific volume in time. Time tap elapsing from the moment of introduction of fuel into the combustion chamber to completion of the conversion process is called conversion time,

SL ae

b)

Fig. 3.8. Conversion curves: a) along the length of the chamber. b) in time,

It is obvious that the volume of the combustion chamber should be such that time of stay of the admitted components will equal the conversion time, With a smaller volume combustion efficiency will worsen, increase of volume is inexpedient, since it will lead to an increase in dimensicns and weight of the chamber without essential improvement of combustion efficiency.

Time of stay of fuel and products of combustion in the chamber

у, t ° oa и (3.1) мпег'� У, — volume of the combustion chamber; G — fuel consumption; Yep — average specific volume of fuel and products of combustion,

If conversion time is equal to time of stay, the value of average specific volume can be determined as the average for time of stay, 1.е.,

ие

(3.2) Since the value wp 1s difficult to determine immediately then for appraisal of time of stay usually there is used certain conditional

time of stay referred to specific volume of products of full combustion Vp (with a given ratio of components):

a (3.3)

100

>>

For simplification of analysis frequently the conversion curve conditionally is replaced by broken line abcd (Fig. 3.8b), i.e., it is assumed that the process of conversion occurs instantly at the time Ts. The location of the position of the vertical line of instantaneous conversion be is determined from the condition of equality of area under the conversion curve 1-2 and the area under section cd. Thus, conversion time is split into two components:

+ (3.4)

where ts delay time of conversion, Time To according to condition of construction of broken line abcd 1s determined from the equation

(ое (3.5)

whence by formula (3.2), considering that 1+ = tnp»

not. (3.6)

Comparing expressions (2.6), (3.1), and (3.3), we will obtain тг =

" Тусд. Putting + = thp and t, = Tyoq in expression (3.4), we obtain

т, (3.7)

i.e., time of stay in the combustion chamber can be presented as the sum of delay time of conversion and conditional time of stay.

Delay time of conversion Tz depends on the kind of fuel, design

parameters of the mixing devices, and on parameters of operation of the combustion chamber,

During operation of the Chamber of a ZhRD in a steady state, delay time of conversion is determined primarily by temperature in

the combustion chamber, and also by conditions of atomization and heat supply to the injected fuel,

101

аб алена

Value of conditional time of stay tyc, 18 one and a half times less than actual. For chambers of a ZhRD, value Туса is within limits of 0,0015-0.005 в. Conditional time of stay (or simply time of stay) is an important parameter of the combustion chamber and is used for determination of its volume (see section 5.2).

The process of conversion in the combustion chamber of a ZhRD is characterized by intense heat emission, Heat release rate of the volume of the combustion chamber of a ZnRD is 100 times higher than the heat release rate of a boiler furnace and 10 times higher than the heat release rate of the chamber of a jet engine.

As a result, in the combustion chamber of a ZhRD there takes place an intense heat supply to the admitted fuel, which leads to considerable reduction in time of evaporation and the entire time of carburetion and ensures rapid evaporation of even comparatively large drops of fuel.

In connection with this, fineness of atomization in a ZnRD has less effect on fullness of conversion than in ordinary thermal machines, This permits producing atomization of fuel with comparatively ил: pressure drops at the sous 3-15 ket/en® (0.29-1.47 М/И?) аз оррозей te 50-100 каг /ст? (\.95-9.81 м/т ) in jet engine and 200-1000 каг /ст? (19.52-98.1 MN/m?) in diesel engines. Decrease in pressure drop at the injectors up to 3-15 kgf /em® (0.29- 1.47 mN/m? ) permits decreasing power, and consequently also the weight of the supply system.

3.2 Spray Injectors

Fuel supply to the combustion chamber is carried out by injectors. The basic requirement for injectors is to ensure the finest possible and most uniform spray of fuel with sufficiently low pressure drop at the injector.

Usually in a ZhRD there are two separate basic forms of injectors: spray and centrifugal. There are also cases of application of slot

injectors which can be considered as a variety of jet injectors

102

having not a round, but a siot-like form of outlet. Injectors are possible which combine both forms of injectors.

The spray injector consists of a hole in the heat of the engine chamber connecting the fuel or oxidizer cavity with the combustion chamber. Different forms of spray injectors are shown schematically in Fig. 3.9.

aloe? А

„) А

Fig. 3.9. Spray injectors: 1-— drillings for supply of components; 2 — concentric annular channels,

Basic advantages of spray injectors are first, simplicity of manufacture and secondly, the great carrying capacity of the head

with the spray injectors,

Carrying capacity of the head we will call the quantity of fuel

passed through a unit of surface of the bottom of the head under a specified pressure drop.

A spray injector occupies less space than < centrifugal. ‘This permits piacing a greater number of spray injectors on a unit of surface of head. Furthermore, discharge coefficient of spray injectors is 2.5-3 times greater than the discharge coefficient of swirlers.

As a result, with the same pressure drop, spray injectors permit ensuring a large flow rate through a unit of the surface of the

bottom of the head, i.e., they have great carrying capacity.

However, a major deficiency of spray injectors is their relatively

large range and small angle of atomization (10-20°); fineness of

103

atomization of jet injectors is less than that of centrifugal. Furthermore, due to the channels for supply of the component to the spray injectors, the head frequently is made relatively heavy

(Fig. 3.9b, c, @).

A larger angle of atomization and better splitting of drops can be obtained if the injectors are set in such a manner that their sprays cross (for injectors with crossing sprays, see Fig. 3.12). Thus, as a result of collision of components there occurs splittinz of drops, and the angle of atomization is increased up to 60-100°, but the passing ability of such injectors decreases somewhat. A block of injectors with crossing sprays can consist of two, three, and even four spray injectors, In Fig. 3.24, 3.2%, 3.27 there are diagrams and photographs of various heads with spray injectors,

Design of Spray Injectors

Spray injector are designed on the basis of the following considerations. As is known, theoretical exit velocity of an 1псот- pressible liquid from an opening

ou, (5.8)

where a6 м (Py - p) — pressure drop, y — density of the liquid,

Flow rate of a liquid through an injector is determined by the equation of flow rate

6,=po/.y. ( 3 ° 9 )

where Г. - area of cross section of hole in m°; u — discharge

coefficient, considering narrowing of the spray and decrease of actual flow rate as compared to theoretical.

Substituting in equation (3.9) the value w from eq ality (3.8), we will obtain

б.у. УЗ, (3.10)

104

from which

fom peo. (3.11)

Flow rate of a component through one injector Gy can be determined, knowing the full flow rate of the component and number of injectors n. If it is assumed that flow rate through all the injectors {8s identical, then

Qn. (3.12)

However, practically rarely does it occur that all the injectors of components have identical flow rate. Usually there are several groups of injectors of one component with various flow rate, and sometimes various construction,

Density of the component is known, Pressure drop at the injector Op is usually given within limits of 3-15 kgt/em> (0.29-4.7 MN/m ) ; to create a large pressure drop is unprofitable since this requires accordingly a large supply pressure, Гесгеазе Spy below 3 ket/em® is irrational since besides it strongly worsens atomization and carburetion, and also increases the possibility of appearance of low-frequency oscillations,

Discharge coefficient u is assigned in accordance with the dimensions of the drilling of the spray injector,

When t/a, = 0,5-1 flow rate decreases by means of narrowing of the spray. In this case (Fig. 3.10a) и = 0.60-0.65. When Иа, = = 2-3 narrowing of the spray also occurs, but pressure in the narrow section 1-1 (Fig. 3.10b) due to negative pressure is less than in the first case, therefore velocity along the narrow section is higher, Thus flow rate of the liquid increases in spite of the narrowing of the spray, Therefore when t/4, = 2-3, value of the discharge coefficient is higher and amounts to yu = 0,75-0.85, Using an injector with large t/a, ratio is inexpedient since thereby there will be increased loss due to friction. Besides, the ratio t/a,»

105

although to a lesser degree, still a number of factors have an effect on discharge coefficient us

a) geometry of the injectors; beveling or rounding cf the inlet edges increases the value of и;

b) pressure drop at the injector АРф; with growth of Ape there can occur separation of flow from the walls of the nozzle of the injector, which with a Given geometry will lead to worsening uw. In Fig. 3.11 there is an example of the dependence of uy on Py with various geometry of the injector, rom which it is evident that with very slight beveling or absence of it in the region App =2-3 КЕГ/ст? (0.19-0.29 МИ т? ) Sharply drops as a result of separation of flow from the walls due to poor exit conditions, Increasing the depth of face e permits avoiding breakaway in a large range of Pps

c) temperature of component, With temperature rise the value of drop oP with which separation o: flow from walls can occur drops;

d) pressure in chamber Po (counterpressure), Decrease of

counterpressure to 5-6 kgf/cm (0,49-0,58 MN/m@) can lead to breakewey of the spray from the walls, and consequently also to a decrease of value о? п;

e) quality of surface of the opening. Various burrs on the edges and much roughness of walls of the opening can lead to considerable reduction of ц,

Beforehand it is impossible to exactly consider the effect of all these factors, therefore during adjustment of construction there always are conducted hydraulic channels for more precise definition of parameters of the injectors, The diameter of the openings of injectors for liquid qd, is usually made within limits of 0.8-2.5 mm,

With a diameter of drilling less than 0.8 mm the openings are easily obstructed. When dc > 2.5 mm, atomization of the component worsens, since the stream of the Component is made too powerful and breaks up into drops poorly,

106

dotmal eutflen

Fig. 3.10, Outflow of component from spray injector,

ij

e270 eee 9.0 т Ape ta /ea*

Fig. 3.11. Dependence of и оп Рф апа on shape of the injector,

Re

Jet Injectors for Generating Gas

During operation of the propulsion system according to a closed diagram (see Chap. VIII) part of the fuel burns in @ gas generator, The forming products of combustion are used to drive the turbine of a turbopump unit and then enter the combustion chamber at high temperature (800-1000°C), For Supplying these gases to the chamber it is expedient to use spray injectors (see Fig. 3.9e). Initially, for calculation of these injectors there will be the equation of flow rate

CoP Of Yun: (3.13)

107

here »p = 0.7-0.85; Yeux — density of products of combustion at outlet pressure from nozzle Рвых = Po}

Van ™ Veo (=). (2.14)

where Рвх and Ypy re pressure and density of gas in front of the injector.

Since pressure drop at the injector Pax -Ppyy &8 compared to pressure in chamber Pos equal to several tens of atmospheres, there 1s little, outflow of gas through the injector will be subcritical with exit velocity

ot! te Aral (tm) |. (2.15)

where R an4 T sx gas constant and temperature of gas in front of the injector,

Placing expression (3.14), (3.15) in equation (3.43), we will obtain calculation expression for determination of fi?

I= (3.16)

ети [1-5 "|

Injectors with Intersecting Sprays

As we noted above, the use of injectors with intersecting Sprays increases the angle of atomization and improves splitting of drops, which leads to an increase o2 chamber efficiency, In the Unites States such injectors first appeared and were successfully used in a ZnRD with nitric acia fuel, Diverse variants of the

diagrams of injectors with intersecting sprays are represented in Fig. 3.12,

108

Reflector

с)

Fig, 3.12, Blocks of injectors with intersecting sprays: a) intersection of sprays of one component; b) intersection of several sprays of two components; c) injector with a "sprayer"; 0 — oxidizer; [— fuel,

®

It is possible to separate three basic forms of injectors intersecting sprays:

with

a) injectors, ensuring paired intersection of Bprays of one

component (Fig. 3.12a).

b) injectors ensuring intersection of sprays of both components

(Fig. 3.12b). Thus in dependence on the ratio of flow rates o and oxidizer (yv) on one fuel injector there can be fixed one, three, and even four injectors for supply of oxidizer, With intersection of two and more streams of oxidizer with one of propellant, for best atomization it is recommended that exit v of combustible exceed exit velocity of oxidizer by 50-604;

109

f fuel two,

elocity

c) injectors with mixing screen (reflector) (Fig. 3.12c). In these injectors, best splitting is ensured by means of collision of stream and wall.

With intersection of two streams, angle of inclination of the resultant stream can be determined, proceeding from the following considerations (Fig. 3.13).

Fig. 3.13. To calculate angles of slop of walls (or openings) in blocks of iniectors,

Let us designate G5» Ops and �>>, respectively as angles of inelination of streams of oxidizer, combustible, and that stream of

fuel formed after collision; m,, Mp, and m, and Wo, Wps and Ws, respectively mass flow rates and velocities of oxidizer, combustible, and fuel. If one were to consider that the momentum of streams before and after their collision remains constant, then projections on a horizontal and vertical axis can be described by the following

equations:

MW, COS Gy} MW, COSE, = MxWy COS Hs: (3.17) т.о, ще, — м, ав, = тир На. (3.18)

Dividing equation (3.18) by equation (3.17), we obtain ап equation of angle of inclination of the resultant stream of fuel

oo мов. ме, — мо, чае. cm 0.2.600 6, + те�&е 06 в ° (3.19)

Since frequently intersection of streams is ensured by corresponding

110

location of surfaces of walls to which the axis of holes are perpendicular, then obviously, angles a, and ар determine slope of walls, In most cases it is desirable to select angles of inclination of walls, flow rates of components and velocities (1.е., 42%) in such

a@ manner that direction of the resultant stream will coincide with axial direction, 1.е., Os = О. Then

M.@, sing, = тю, те, (2.20) ог биг, зта' = Слеза, (3.21)

It must be noted thet manufacture with the necessary degree of accuracy of a large number of injectors with intersecting streams

18 no less complicated a problem than manufacture and installation of separate swirlers,

Example of calculation of Spray injectors

To determine the dimensions of spray injectors and angle of inclination of oxidizer injectors with an = 30°; a> = 0;

м en? 0.90 rae ¥, = 0,8 gen? Oy, =73 o/s; pg, = 7 nat /en® (0,68 rove"), Vo = 1,8 e/en’,

solution, Considering 1/d = 3, we take for oth injectors, discharge coefficient ци = 0,8,

We determine dimensions of injectors, By formula (3.11):

№ О © 177-1 ot 73-10-08

- Ye AVERETT TET = 20 1-4 out

Considering coefficient of velocity to equal discharge coefficient и, we determine actual exit velocity:

111

Уч му 3.0.61 �"" - 94 a/e; > У 1, Go 9.85/ 9.0.0) a 4.2 af

Since а; = 0, when a, we will define by equation (3.21):

"please Fe stn, = SEES vay 0,377.

whence angle of inclination of oxidizer injector о, = 225101. 3.3. Centrifugal Injectors

Centrifugal is the name applied to an injector in which there is artificially created a swirl of the stream of liquid passed through it. After exit of the liquid from the nozzle under action of centrifugal forces there will be formed a thin coneshaped liquid sheet of component which rapidly breaks up into drops (see Fig, 3,3b). Therefore, the diagram of flow intensity of a centrifugal injector has two peaks (see Fig. 3.3d).

Centrifugal injectors have a wide and comparatively short Spray cone, The spray of centrifugal injectors {s finer than for jet. This all leads to a decrease of zones of atomization and evaporation. However, a disadvantage of centrifugal injectors is their great structural complexity and smaller carrying capacity as compared to spray injectors,

According to the method of obtaining the flow swirl of component, centrifugal injectors are divided into tangential (Fig. 3.14a, b and 3.15а, b, c) and injectors with swirl vane (or screw (Fig, 3.44¢ апа 3.15е).

In a centrifugal tangential injector, the liquid enters the cavity of the injector through one or several inlets, the axes of which are perpendicular to the axis of the injector but do not intersect it. Sometimes the opening is made at er acute angle to the axis of the injector, As a result, the liquid obtains swirl with respect to the axis of the injector,

112

Fig. 3.14. Diagrams of centrifugal injec- tors: a) tangential closed; b) tangential open; 1 — liquid inlet; 2 — swirl vane (screw); 3 — swirl chamber,

J

4 AY Sf

A И |

№

2 Ld LO ЛТЗ UW

4) Pig. 3.15. Centrifugal injectors: a, b, c) tangential; d, e) screw; 1 — body; 2— screw; 3 — nozzle,

113

There are open and closed tangential centrifugal injectors, For closed tangential injectors (see Fig. 3.14a), radius of nozzle Г. is less than radius of the swirl chamber R, For open injectors, radius of nozzle is equal to radius of swirl chamber (see Fig. 3.44b), In an injector with a swirl vane (see Fig. 3.14c) swirl is created by means of a special swirl vane (screw) which has a screw thread on the external surface, Moving by screw thread, the liquid obtains a swirl with respect to the axis of the injector, Let us consider the operation of a centrifugal injector [1].

Operation of an Injector

In a centrifugal tangential injector (Fig. 3.16), the liquid enters the cavity of the injector through an inlet having radius with velocity Way This opening is located so that its axis is tangent to the circumference of radius R,y with the center located on the axis of the injector nozzle, Thanks to this inlet, liquid passes through the cavity into the nozzle of the injector, revolving. Let us consider a stream of liquid which, moving along the injector, gets to the nozzle at distance r from its axis,

Tax

Fig. 3.15. Motion of liquid in a swirler; 1 — liquid; 2 — gas swirl; 3 — "active" section,

If we disregard the action of forces of friction, then angular momentum of any liquid particle with respect to the axis of injector should maintain constant value all the way from entrance into the injector to exit from nozzle of it, i.e.,

114

®„А„ =, (3. 22)

where Wy ~ peripheral velocity of motion of a particle of liquid in the nozzle at distance r from the nozzle axis,

Since one may assume that on the average for all streams (with Slight error due to change of value ly ) the momentum of the liquid, achieved by it in the inlet is the same, then velocity м, depends on radius r, on which this stream gets into the nozzle;

o,— San, (3.23) Disregarding the insignificant difference of levels of location

of inlet and nozzle openings, pressure in the stream of liquid can be determined by the Bernoulli equation:

o oe Ors 4, et, 2.4 Te Se ‚24 + ты + const, (3.24)

where Pax ~ Pressure of liquid in the inlet; Wax — entrance velocity of the liquid into the injector; Wy — tangential component of velocity

of liquid at exit from the injector; W, —~ axial component of velocity of liquid at exit from the injector,

Designating total pressure drop at the injector through SP» and expressing it through pressure H, we will obtain

> И 1+ te conet, (3.25)

After that, from equation (3.24) we will obtain

fun-(242), (3.25)

From equations (3.23) and (3.2) it 4s clear that with Г > 0 there will be Ww, 7, i.e., pressure of liquid on the axes of the injector should have infinitely large negative value. This is impossible

for liquids since a liquid in general does not withstand negative stresses, 1.e., does not work on extension,

115

In reality, in the injector the following occurs. As the liquid approaches, the axis of the injector, velocity Wy, Will be increased, and pressure p will drop, but only until it becomes equal to ambient pressure in which outflow occurs (with injection into the chamber — pressure in chamber). Further decrease of pressure in the central region of flow is impossible, Since by its only base this region emerges through the nozzle into the environment, the central part of the injector will not be filled with liquid. In it there will be a gus vortex with a pressure equal to ambient pressure (pressure in the chamber), Flow of liquid by the nozzle or the injector will be carried out not through the whole section but only through the annular, the internal radius of which is equal to radius of the gas vortex Tn? and external radius — radius of the nozzle Toe

This section we will call the useful cross section of nozzle of the injector; its area

Лы="�""-"). (3.27) Flow rate through the nozzle of the injector 9 Литтл (1 п) ил, (3.28)

where @— coefficient of useful cross section, It is obvious that

(3.29)

em l—

We will define change of We and Wo by cross section of stream, Let us consider section of stream at the nozzle exit section of the injector (Fig. 3.17). Let us separate in useful cross section at distance r from the axis, annular element dr, According to the d'Alembert principle, difference of pressures on the surface of the annular element Ap is balanced by centrifugal force, For an individual element, the equation of equilibrium will have the form

арт вл, (3.30)

where dm — mass of individual element

116

dmm—. gp, #

(2.21) According to equation (3.22) it 18 possible to write OF PO mn (3.32) where Wum ~ tangential velocity of liquid with r «= Ги. Hence я мы — "р dr Ри, (2.33) After substitution of expressions (3.31) and (3.33) in equation (3.30) we will obtain dp= —te,¢e,, (3.34) and after integration 2 с 3.35) Yat . ( й

Let us find constant С. Py — Pressure, excess above border of swirl and liquid p

With w= Wom it will be p = P,» where pressure in vortex Po. Obviously, m = 0» whence

on

�"".

Then equation (3.35) will have the form

2 т 1% №’ Вы

Comparing expression (3.36) and (2.26), we will obtain

ian, (3.37)

1.e., axial component of velocit

y of liquid in useful Cross section of the nozzle of the injector w

5 does not depend on r and ig constant over the whole section, i.e,, @ = const, (3.38) 117

aaa ie

»)

Fig. 3.17. For determination of forces having effect on the annular element: a) useful cross section; b, c) change of Wa» W), and 2a by useful cross

section.

We will define change Wy by section, Equation of constancy of flow rate for inlet and for section on nozzle exit section of the injector

C,=0,/ Vow rly aw, ar! y, (3.39)

Placing from equation (3,22) the value Wpx in equation (3.39), we obtain

ви “+. (2.40)

Diagrams of change of We and Wy by useful cross section are represented in Fig. 3.17b.

Geometric Characteristics of an Injector

Into expression (3.40) there enters complex Refelfi,, connecting the basic dimensions of an injector, This complex is usually designated as A and called the geometric characteristic of a centrifugal injector,

Ач. (3.41)

As we will see further on, the geometric characteristic is the most important parameter of centrifugal injector,

118

me

=

. 9: =. А 4 '

In the analysis made, we determined the geometric characteristic

A for a tangential injector with a Single inlet, Conducting analogous

Computations, it ‘g easy to find the expression of geometric character- -istic for other types of Centrifugal injectors,

Thus, in general, for a tangential

anjector with several inlets inclined at an angle to the axis

of the injector;

Aw tx sin}, (3.42)

where 1 — number of inlets;

B — angle between directions of axes of inlets and nozzle of injector

For an open injector (see Fig, 3.146), в1псе ro = Rey,

Am Ziad (3.43)

For a screw injector (see Fig, 3.15e)

& Aun Sn sing, (3.44)

where Rpy — average radius of screw c angle of ascent of the Spiral; f channel; i — number of approache vane (number of channels),

hannel of the swirl vane; p — 1 —~ rea of cross section of one 8 of the screw thread of the swirl

(3.45)

Where r = Ги? velocity Ww Маш @nd since telrm=1/Vi-e [see (3.29) ] tangential velocity on border of whirl

Ты, = A (3.46 ) Where r = Го

119

@, .-=OgA. (3.47)

Discharge Coefficient of an Injector

Using the obtained dependences, we will define flow rate through the injector Gg. .

З1псе Н = Apg/y (3.25), then from equations (3.37) and (3.45) we . will obtain

(3.48)

Then according to equation (3.39) “УЕ (3.49) " 1-

ры! (3.50) ИТ

and to substitute fy = mre, then we will obtain an expression identical to the equation of flow rate through a spray injector (3.10):

If one were to designate

Gy=pf.y Beary. (3.51)

where ц — discharge coefficient of a centrifugal injector,

From expression (3.50) it is clear that the discharge coefficient и depends on the coefficient of useful cross Section 9, 1.e., on area of useful cross section fy+ It is obvious that with fy > 0, discharge coefficient and flow rate will approach zero, However, even a strong increase of fy also will lead to & decrease of i, inasmuch as with large fy (1.e., with very small radii of whirl Tn) axial velocity Wa decreases considerably since a large part of the

120

+=

pressure will be expended on creation of high peripheral velocity Wye Consequently, there is some optimum value of coefficient @ with which value « will be the greatest, According to the principle of maximum flow rate known in hydraulics during flow of a Component through the injector there should be established such value of useful cross section к, with which fiow rate through the injector will be the greatest. Thus, for each Given value of A with change of useful

Cross section, 1.е,, coefficient » is its maximum value of discharge coefficient 1. For its determination the derivative du/dp will equate

to zero; | 1 - —.. i -

(+)

whence

Consequently, the greatest discharge coefficient will be during

condition Та — А-З. (3.54)

Dependence of 9 on A with a condition of maximum flow rate is shown in Fig, 3,18, Substituting expression (3.54) in equation (3.50), we will obtain a connection between и and o;

А = (2.55)

after which it is Possible to construct the dependence of on geometric characteristic A (Fig. 3.48),

Thus, the value of discharge coefficient u is determined by the value of geometric characteristic of a centrifugal injector A, Limits of change of y' are easy to find from expressions (3.54) and (3.55): мПеге А -> О there will Бе ф>> 1 and и => 1; where Amo there will be 9—~ 0 and u—0; with all remaining values of A1> > O, and the

122

values of » according to the relationship (3.55) will always be smaller than 9.

wy Г ie? Fig. 3.18. Change of 1, ф, and 2a depending on Г geometricouch character- ? istic A, гг) $

Knowing п, we can determine the necessary area of nozzle of an injector from equation (3.51):

hn ete. (3.56)

Geometric characteristic A is connected also with angle of flame of atomization 2a.

As can be seen from Fig. 3.17c, the angle of atomization a is inconstant for useful cross section and for peripheral streams decreases in conformity with decrease of Wye Therefore we will determine the average angle of flame of atomization for a mean value of radius of useful cross section

rg ete. (3.57) It is obvious that gem”. (3.58)

Since м, ав М ахГс/Гер (3.22) than taking into account expressions

(3.29), (3.47) and (3.57) we will obtain

ta— Fee Tel — = 1 , (2.59)

and after substitution of equality (3.59) in formula (3.58)

122

ЗУТ( — Substituting in the last expression instead of » the value of А connected with @ equation (3.54), we will find the dependence of average angle of flame of atomization 2a on geometric characteristic A (see Fig. 3.18),

Effect of Viscosity on Operation of an Injector

Above we have given an analysis of the operation of a centrifugal injector with Supply of an ideal liquid, With supply of an actual liquid possessing viscosity, the presence of forces of friction leads to a change of discharge coefficient and angle of atomization 2a, Friction leads to a decrease of angular momentum of fluid flow along the length of the injector. Due to this, swirl of flow, i.@., Wy decreases, which leads to a decrease in radius of gas whirl Га’ decrease of Tn Signifies increase of useful cross section of injector nozzle, i.e., increase of discharge coefficient . and although due to losses of pressure by friction, flow rate decreases, on the whole, the effect of decrease of intensity of swirl turns out to be stronger than the effect losses of pressure by friction, Therefore, discharge coefficient д for an actual liquid is greater than for ideal,

Angle of atomization 2a thus decreases, since tangential component of speed Wy decreases, Thus, viscosity leads to a decrease of angle of atomization 2a, and paradoxically, at first glance, the result — increase of discharge coefficient of a centrifugal injector ц. AS can be seen from the Graph in Fig, 3.18, we would come to the Same result with a decrease in value of geometric characteristic A,

Therefore, a centrifugal injector having geometric characteristic

A, with supply of actual liquid can be calculated with the use of the S0-called equivalent characteristic of injector Аз» smaller than A;

where Ky < 1,

123

According to L. A. Klyachko [27]

‘| and А, = 4 (3.63)

where i — number of inlet channels; B — dimensionless geometric parameter of injector; Bo = Ват — for tangential injector;

Bo = Ввхт/ 24 — Гог всгем 1п]есфог; A — coefficient of friction, determined during conditions of entrance into the injector by the formula:

ghar 2. (3.64) Where Magn, (3.65)

where anp — given diameter of inlet channels, For an injector with i number entrance channels by overall area fpyi

„У “В а, УТ.

Entrance velocity of component

�"" м `

Substituting the expressions for апр and w,, in equation (3.65), and considering that v = gu/y, we will obtain a calculation formula for determination of Repy:

Reg aor (3.66)

124

o

Obviously, for an ideal liquid, the coefficient of friction 4 = O and according to formula (3.63)

А.=А.

During calculation of injector formla (3.63) it is more Convenient to reduce to the form (with sin B = 1)

и. (3.67 Wea 5 Вы (ит)

Effect of Design Parameters of an Injector

We will examine the effect of different design parameters on the operation of a centrifugal injector,

Increase of height (length) of injector h (see Fig 3,15) increaser the effect of forces of viscosity which along with increase of ы and decrease of angle 2a leads also to increase of losses of pressure in the injector, i.e,, to certain impairment of atomization, Therefore it is fully sufficient with a screw injector to make the height of the screw not more than 1/4-1/3 pitch of turn, and with a tangential injector — equal Rpy, although in practice, due to purely structural

considerations frequently it is necessary to accept considerably higher injectors,

Ratio Ввх/ Го is usually selected within limits from 1 (for an open injector) to 2,5,

As experimental data show, with values Авх/ То < 2.5, experimental values of discharge coefficient Нэксп &re obtained less than the theoretical obtained on the Graph in Fig, 3.18. In Fig, 3.419 there is a@ graph of dependence of ration РэкспИН on ratio Ввх/Го [27].

Change of’ ratio 1</а, (вее Fig. 3.16) essentially does not effect discharge coefficient м. However, with growth of Иа angle

of atomization 2a decreases, Therefore usually 1/4, = 0.25-1.

125

зи, ыы i

o Fig. 3.19. Dependence of Hexcn/+ On Rpy/r,.

sy

Thickness of wall of the injector is selected from condition 1вх/@вх = 1.5-3. With values 1,,/d,, < 1.5 the trngentialness of inlet is disturbed (Fig. 3.20) which leads to a decrease of angular momentum with respect to the axis of the injector, Increasing lax /dpx is also inexpedient since it leads to unnecessary pressure losses by friction in the inlets,

/ Fig, 3.20, Change of direction of

inlet of liquid with smell values

™ of lpy/dgx. 1 — direction with tangential inlet, 2 — actual a direction,

The number of entrance channels i (or approaches of the screw) usually is taken as equal to 2-4, Increase in the number of entrance channels, obviously, improves distribution of flow intensity along the circumference of the flame.

Effect of Preheating the Component

Since the combustion chamber of a ZnRD is usually cooled by one of the components (and sometimes by two components), the cooling component proceeds to injector heated and thus, frequently up to a temperature close to the boiling point or even equal to it. Therefore the question about effect of temperature of preheating of component on operation of a centrifugal injector has practical value, If the temperature of liquid, entering the injector is such that the pressure of saturated vapors (vapor pressure ) Ра does not exceed pressure in the gas vortex (equal to pressure in combustion chamber Po), then there will not be essential changes in operation of the injector.

126

on

It is necessary only to consider change of density of component due

to temperature. However if Ра > Po» “hen boiling of component occurs in the injector and from the nozzle o. the injector into the combustion chamber there proceeds not liquid but a vaporliquid mixture, Naturally

this affects the operation of the injector and, primarily the discharge coefficient of injector м.

A sufficiently detailed analysis of operation of an injector during flow of hetted or boiling liquid is given 1п [58], [61]. In [58] it is proposed that the effect of preheating on discharge coefficient be considered by introduction of a discharge coefficient

of heated liquid +:

Pos ВУ д, (3.58) where — discharge coefficient during outflow of cold 119914; р. — pressure in front of injector, It must be considered that with Ps» close to Py: formula (3.68) gives a decreased value of HW, and with р; = № there is obtained MW, = O which does not correspond to experi- mental data. Therefore, with values of Ps Close to Py one should take somewhat oversized Values и, &8 Compared to those obtained by formual (3.68). When more exact analytic determination of и. 18 necessary it is possible to use the results in (62), where there is a more precise but considerably more complicated dependence for determination of Hye In Fig. 3,21 there is a comparison of values

of и, obtained from experiment by formula (3.68) and the formulas in [62]. |

} ») Fig. 3,24, Change of discharge coeffi- cient Me depending upon temperature of

119414: a) water; Py = 30 kgt/om* (=3 Mi/n*); b) diesel fuel; Py = 20 ket/en?

2 (w2 MN/n™); 1 — disregarding evaporation of liquid in the vortex; 2 — according to formula (3.68); 3 — according to formulas in (62),

127

Calculation of a Centrifugal Injector

In the calculation of a centrifugal injector we assume as known the flow rate of the component through the injector Gy, and its physical properties. Thus the following order of calculation of a centrifugal injector is used.

1. Depending upon operating conditions of the injector there is selected angle ae and drop SPH a limits of 2a = 30- -120°; dpg = 3-15 ket /om® (0.29-1.47 М/И? ). The most widely used values of angle 2u lie within limits of 90- 120°, however under certain specific specific conditions injectors with smaller angles can be required, Limits of pressure drop Pp are determined under the same considerations as for a spray injector.

2. Knowing angle 2a, from the graph сГ Fig. 3.18 we determine geometric characteristic A and discharge coefficient i.

3. We determine area of section of the nozzle of the injector and diameter of the nozzle. According to formula (3.56)

= wal ad, he же. whence 4, Vt

4. From constructive considerations, considering the effect of different parameters on operation of the injector, we select the number of inlets 1 and the "arm" swirl Ryy/r,. Knowing 1 and R5y/T,

we determine any =.

5. Knowing ray, from formulas (3.66) and (3.64) we determine coefficient of friction i,

6. With.r,, Roy, Tgg, and ’ now known by formula (3.67) we determine

Ae

т.м

126

~

Se _

If the obtained value Ay differs from geometric characteristic A by

no more than 5%, then this calculation is finished; values Гс› Rox;

and Ty, Of first approximation are taken for final and the remaining dimensions of the injector are determined. If the divergence cf A

and A, is large, then the calculation is conducted repeatedly. Taking for the base the obtained value Ag according to the graphs of Fig. 3,18

we determine a new п (already taking into account viscosity), and then new values гс, Ray: and Psy:

According to the new values Tos Ray: and Гвх we determine the equivalent geometric characteristic of the second approximation atl and compare it with Ag of first approximation. If large divergence is again obtained than a third approximaticn is made, But usually

by the second approximation act is sufficiently close to A, of first approximation,

7. According to values obtained during the last approximation

for a Арх, апа Tay the remaining dimensions of the injector are determined (see Fig. 3.16):

6-36; 1 =0.251)6; АА,

Radius of swirl chamber Rs = Ray + Грх. ЦП, during calculation of the injector it appears necessary to consider preheating of the component passed to the injector, then, according to formula (3.58) we correct the value of the discharge coefficient obtained in point 2, and conduct further calculation with a new value of discharge coefficient pe

Example of calculation of a centrifugal injector

To calculate, taking into account viscosity, a tangential centrifugal injector for supplying nitric acid at its temperature of 268°K, Flow rate of acid through the injector Gy = 68 в/в;

density THMO = 1.51 g/cm; viscosity g u = 0,981 x 41072 kgf/m x в (0.964 х 107? нх в).

Solution. We are assigned angle of atomization 2a = 400° and pressure drop dp, = 8 kgt/en* (0.78 MN/m*) , In accurdance with the

129

ee. eee

Фит

graph in Fig. 3.18 for angle 2a we determine geometric characteristic A and discharge coefficient ui:

Am42, p-0,16.

We determine the dimensions of the nozzle of the injectcr:

0,008 hea SVERRT ony TRE TIT = 8,72-10-2 eat,

eo у 3a" Ииал-ам Ш; г. = 1,67 ди.

Since we are conducting calculation taking into account viscosity, then correction of dimensions obtained in first approximation (rounding to integers) for the time being will not be made.

We take Ry /*e = 2.5 and i= 2, Then

Reg = 2,86 = 3,5.1,67 = 4,16 лм; Ош == 05/2 = 34 г[сок; rae V a7) SET а ил; Ци = 1A a

We determine coefficient of friction 4. By formula (3.66)

4-0,008 Mam a aa" Teresa le

By formula (3.64) we determine 4 = 0.0331. By formula (3.67) we determine

В”, _ 4,10-1.07 a= i = 0,083) wie: ats АА. —^) — 2.0.911 + So 418 4.18— 1.67)

The obtained value A, differs from A by 9.5%, therefore we calculate the injector in second approximation according to geometric character- istic Ag.

According to the graph in Fig. 3.18 when A, = 3.83 we find the discharge coefficient, п = 0.17 taking viscosity into account.

130

We determine the dimensions of the nozzle of the injector, By formula (3.56)

= УРАГАН см enV iy itn an,

we take d. = 3.2 mm; Ге = 1.6 mm; Авх/ Ге = 2.5, Then

* Reg 3.8-71 3.5.1.6 #4 war: aE Bm

We take ry, = 0.9 mm; dp, = 1.8 mm, By formulas (3.65) and (3.64)

we determine Regy and coefficient of friction with definitized dimensions of the injector: Reg, = 34,400, Since Regy changed insignificantly, the coefficient of friction, calculated by formula (3.64), remains equal to A = 0.0331.

We determine value Ag. according to second approximation:

20.04 AS a 1,9)

Value ait differs from A, by 5% which is within permissible limits, Therefore we consider the values Tor Kgy,

and Spy defined in second approximation as final,

Let us define the remaining dimensions of the injector:

ди = 1.66 > 1,5:1,0008,7 дл; и 0,56, 0,6-3,3 со 1,6 дл;

height of injector h = Ray = 4 mm; radius

of swirl chamber Ry = Rey + + lex a4 + 0,9 = 4.9 mm,

121

3.4. Two-Component Injectors

Diagrams of Two-Component Injectors. Advantages and Disadvantages

Along with single-component injectors, use is also made of two- component centrifugal injectors, basic diagrams of which are presented in Mg. 3.22. Two f: rms of two-component injectors are distinguished:

A — with internal mixing (emulsion).

B — with external mixing.

In two-component injectors witi: internal mixing, the mixing of components occurs in the injector just prior to their entry into the combustion chamber. Inside the injector both components will form an emulsion mixture which enters the combustion chamber, therefore these injectors are frequently called emulsion. Such injectors frequently are used expediently with supply of nonhypergolic components, since with hypergolic components, the combustion reaction can begin before the emulsion of component emerge from the injector, which will lead to destruction of the injector and the combustion chamber,

же ий

Pig. Diagrams o ме. injectors: A &: Aon ‘internal mixing (emulsion); By with

ernal mixing; a, 4, e) closed injectors; :] open injectors; b) иен injector; c ot injector.

132

In two-component injectors with external mixing, the mixing of components occurs after they leave the injector, Thus, one may assume that a two-component injector with external mixing essentially consists of a structural block of two single-component injectors

ensuring mixing of components in a prescribed ratio directly at the engine head.

Two-component injectors are not necessarily a combination of two centrifugal injectors, Different structural combinations of centrifugal, jet, and slot, injectors are possible, In Fig. 3.22¢ there is a diagram of a two-component injector with external mixing in which a centrifugal and a slot injector are combined,

Use of two-component injectors permits improving carburetion since basic mixing is ensured of components still in liquid phase, which leads to a more rapid flow of the entire process of combustion, and means that it permits decreasing the volume of combustion chamber required, The carrying capacity of a head with two-component injectors is greater than with single-component centrifugal injectors.

The disadvantages of two-component injectors are first, their great structural complexity and secondly, more rigid thermal conditions of head operation. Since in using two-component injectors the zones of atomization and evaporation are shortened, the flame front approaches the head and the intensity of heat flow from flame front to head increases,

Let us consider procedures for Calculating two-component injectors,

Calculation of Emulsion Injectors

The calculation diagram is shown in Fig. 3.23a. Let us designate 153 153 Mos “pd od Ynd Ge gd @ф.г respectively as number of inlets, inlet velocity, density and flow rates of oxidizer and fuel through

the injector, For community we consider that the openings for supply of fuel are inclined at angle В.

133

=e

Fig. 3.23. For calculation of two-component injectors: a) with internal mixing; b) with external mixing; 1 — emulsion; 2 — opening for introduction of fuel; 3 — internal injector;

4 — external injector.

Let us assume that Ray ~ Ray p= Rayo and that operating pressure drops during supply of fuel and oxidizer are equal; action of forces of friction will be disregarded. Then it 1s possible to write down

wer +. (3.69)

where Dg.o = Pg.r — pressure of oxidizer and combustible in the mixing cavity of the injector,

From equation (3.69) it follows that with psp = ppp

= Va, (3.70) where enh. (3.71)

Fuel consumption through the injector

О *!. (3.72)

Based on the law of conservation of moment of momentum we can write down:

бобы сое + бо, (3.73)

where Ws — tangential component of velocity of mixed flow in the mixing cavity.

ел. =.” (2.74)

where Wy — tangential component of velocity of emulsion at nozzle outlet.

From equation (3.73), taking into account expressions (3.70) and (3.72) we obtain

к-т (3.75) and from formulas (3.74) and (3.75) we determine ae У ©. (2.75)

Let us define density of fuel 7; formed during mixing of fuel and oxidizer in the injector. Since

‘te © then taking into account expressions (3.71) and (3.72) =. (3.78)

We will define peripheral velocity of emulsion in nozzle of injector

My e

135

Analogously to equation (3.39), taking into account dependence (3.72), the equation of flow rate through the injector

бы тии lari, =", (3.79)

where We = axial component of velocity of emulsion in the nozzle,

Then, comparing expressions (3.76), (3.78), and (3.79), we obtain

(v+ymenat)v Rute г Са ae

Designating

ram ee ara |

we obtain ее, | (3.82) where Ag, — geometric characteristic of an emulsion injector. Equations (3.82) and (3.45) are analogous,

Performing conversions analogous to the conversions for a single- component injector we can be convinced that dependence of » and 2a оп Азы 18 the same as for a single-component injector (see Fig. 3,18). Consequently, knowing the fuel (i.e., » and m) and having the radius of oxidizer inlets го and their number 1, we can conduct further calculation of a two-component emulsion injector in that same order as the calculation of single-component injectors, using for this the geometric characteristic of an emulsion injector Ag, instead , of geometric characteristic А, .

Calculation of Two-Component Injectors with ; External Mixing

Calculation of two-component injectors with external mixing (Fig. 3.23) basically is reduced to the calculation of internal

136

and external injectors examined as independent single-component injectors, Thus the radius of Swirl of the external injector rl should be greater than the external radius of the body of the nozzle of the internal injector pil 1.е., ri > ТТ, In the case when

ri < pil for the external injector part of the useful cross section

will be blocked by the body of the internal injector,

Radius of swirl of the external injector ri is easy to determine, knowing the geometric characteristic of the injector, since formulas (3.29) and (3.54) give us a connection between A, coefficient of useful cross section », and the ratio Ту Ге,

Solving jointly equations (3.29) and (3.54), we find for an ideal liquid the connection between т/с and A,

Angle of atomization of an external injector ат can be less than the angle of atomization of an internal injector 2a! (as shown in Fig. 3.23b) and greater. In the first case intersection of the flames of atomization ensures the best mixing, With 2at > 2qrt there is ensured the best protection of the head against burnout (in so doing frequently fuel is fed into the external injector), Concluding the consideration of different types of injectors (spray, centrifugal, mono, and two-component, etc.,), it must be noted that an important Stage in the development of injectors is their hydraulic testing, which usually are conducted with water. These tests permit correcting the calculation discharge coefficients and angles of atomization, and also obtaining the necessary data on mixing and distribution of the component according to chamber section,

3.5. Chamber Heads for the ZhRD

The chamber head of an engine is the main subassembly providing correct carburetion in the combustion chamber. The head should be 80 designed as to ensure stable burning in the chamber, promote smooth ruiup of the éngine, and decrease the impulse of the aftereffect (see section 5.5). The design of the head should provide for

137

necessary placement and reliable attachment of injectors, most convenient feed of components to injectors, and simplest technologi- cally possible attachment of the head to the comp ition chamber.

Types of Heads for the ZhRD The basic types of heads are flat, tent, and spherical.

Flat heads (Fig. 3.24a, b, c) are the most widespread type. The advantage of flat heads is simplicity of construction. Furthermore, flat heads make it possible to ensure homogeneity of velocity field and of concentrations of fuel over the cross section of the combustion chamber. The deficiency of flat heads is their relatively low strength and rigidity. Therefore in the flat heads of large engines it is necessary to provide reinforcing elements, ensuring required strength and rigidity of head.

Fig. 3.24. Diagrams of heads for ZhRD:

a) flat with double bottom; b) flat with drilled holes; c) flat with crossing streams of oxidizer and fuel; d) spherical; е) spherical with precom- bustion chambers; f) tent with central feed of oxidizer; 1 — upper bottom; 2 — middle bottom; 3 — lower bottom; 4 — coolant cavity; 5 — collector rings; 6 — drilled holes; 7 — igniter;

8 — 05 cavity; 9 -

Но annular passages;

10 — valve; 11 - precombustion chambers.

a)

Variant I

ariant II (gas

�'' BY 4)

138

RE 9 ee

In Fig. 3.25 is shown a flat head, working as illustrated in Fig. 2.:'a. Coolant O flows from the coolant passage to the cavity between middle 2 and lower 3 bottoms cf the head, whence through screw injectors 6 it enters the combustion chamber. Component I passes through inlet tube 10 into the cavity between upper 1 and middle 2 bottoms and from there through screw injectors 5 to the combustion chamber. The injectors are attached by means of beading. The head is connected to the combustion chamber by connecting ring 4 and also directly by welding to the inner shell of the chamber 7.

Fig. 3.25. Flat head: 1 — upper bottom; 2 — middle bottom; 3 — lower bottom; 4 — connecting ring; 5 — fuel injector; 6 — oxidizer injector; 7 — inner shell of combustion chamber;

8 — chamber housing; 9 — inlet tube; 10 — tube.

In Figs. 3.26 and 5.3 are shown the extremal appearance and cutaway view of a head with drilled holes, made in accordance with the diagram of Fig. 3.24b.

Fig. 3.26. Flat head with spray injectors.

Spherical heads have found application chiefly in high-thrust engines. The merit of such heads is structural rigidity.

In Figs. 3.24d and 3.27 are shown a diagram and external appearance of the spherical head of the RL-10 oxygen-hydrogen engine.

139

Fig. 3.27. Spherical head with spray injectors.

In Figs. 3.24d and 5.4 are shown a diagram and cutaway view of the spherical head of oxygen-alcohol engine of the A-4 missile.

Tent heads, shaped like tents (see Fig. 3.24e), find application in low- and medium-thrust engines and also as precombustion chambers. The advantages of the tent head are larger surface for placement of injectors than for flat head and good strength properties. The deficiencies of the head are complexity of manufacture ana irregularity of fuel distribution over their cross section. With the tent head formation a "bunch." of atomized fuel is possible. In Fig. 3.28 is shown the precombustion chamber of the head.

Fig. 3.28. Precombustion chamber of tne head of the engine’ 1 —

oxidizer feed tube; 2 — central

spray injector; 3 — upper row of swirlers; 4 — lateral spray injectors; 5, 6 — combined spary injectors and swirlers; 7 — inner shell of precombustion chamber.

140

=o

-.- oe

LE ь

Placement and Mounting of Injectors on Head

Placement of injectors on the head should promote fulfillment of the basic requirements imposed on carburetion while ensuring

reliability and technological effectiveness of design, which basically is reduced to the following.

1. Most equal possibie distribution over the cross section

of the combustion chamber of the ratio of components a and flow intensity r.

2. Least possible inclination toward appearance of unstable burning.

3. Protection of combustion chamber walls from burnout.

4. Protection of combustion-chamber head from the influence of high heat flows from the flame front.

5. Convenience of feed of components.

Investigations have shown that the distribution of ratio and flow intensity of components obtained directly at the head for practi- cal purposes is preserved along the entire combustion chamber and nozzle of the engine. In turn, irregularity of distribution of a and

r over the cross section of a chamber affects specific thrust of an engine Руд (see section 3.6).

Let us consider the basic schemes of injector placement. In engines operating on single-component injectors in order to ensure g00d carburetion uniform alternation of fuel and oxidizer is necessary

injectors. Therefore it is possible to distinguish the following basic patterns for fuel and oxidizer injector placement.

Checkerboard placement. For which fuel and oxidizer injectors are arranged in checkerboard fashion in alternating order (Fig. 3.29a)

The deficiency of this method consists in the fact that the number of fuel injectors is approximately equal to the number of oxidizer

141

oe зы

injectors. Since the mass flow rate of the oxidizer is usually

2-4 times greater than that of fuel, with such placement the

flow rate of each oxidizer injector is considerably greater than that of the fuel injector, which can worsen carburetion, since the powerful stream of oxidizer mixes poorly with the relatively weak fuel spray, knocking it aside.

Honeycomb placement (Fig. 3.29b). Here each fuel injector is surrounded by a group of oxidizer injectors, making it possible

to have a greater oxidizer than fuel injectors. The difference in flow rates of the oxidizer and fuel injectors is less here than

with the checkerboard arrangements, which provided better atomization and mixing of fuel components.

Concentric placement. Here belts of fuel and oxidizer injectors are alternated (Fig. 3.29c). In certain designs such arrangement

simplifies the feed of components to the injectors. An example of concentric location of injectors is the head of the oxygen-hydrogen engine (see Figs. 3.24d and 3.27). Here in concentric belts are

groups of injectors, consisting of two fuel (He) and one oxidizer (0,) injectors each.

Fig. 3.29. Diagram of injector placement: a)

checkerboard; DS honeycomb; c) concentric; x —

fuel injector; o —‘oxidizer injector; e —- fuel injector for creation of near-wall layer.

Two-component injectors can be placed in any arrangement. It 1s necessary only to consider the possibility of appearance of unstable burning and the necessity of protecting the head from burnout.

142

РЕ оне

Ым—�>�—>—=>=“—=>- ee

To ensure conditions least promoting the appearance of unstable burning, both Single- and two-component injectors are sometimes placed in orders, constituting different combinations of the above-mentioned arrangements, and attempts are also made to extend the process of burning somewhat over the length of the combustion chamber. At the end of chamber it is desirable that there be a heterogenecus zone, which damps as far as possible longitudinal high-frequency combustion instability (4). This is attained by a certain alternation of injec- tors with different flow rates and spray angles (1.е., range).

For protection of combustion-chamber walls from burnout a protective near-wall layer, overly rich in fuel, and consequently of lower temperature than the flow core, is created. It is necessary to note that a near-wall layer with a great surplus of oxidizer also would have a temperature lower than that of the flow core and would apparently be a fully satisfactory protective layer, but the danger of appearance of local seats of burning at the wall én the oxidizing medium and burnout of walls due to oxidation of metal is the reason the fuel-enriched near-wall layer is usually created. For this purpose on the head there either is placed a Special peripheral belt of fuel injectors, as in Fig. 3.29 or the outer oxidizer injectors are replaced by fuel injectors.

The peripheral fuel injectors usually have longer range and lower flow rate than the main injectors. The spacing between peripheral ("protective") injectors and their positions are selected in such a way as to ensure uniform thickness of the near-wall layer about the perimeter of the chamber. Local increase in thickness of the near-wall layer leads only to increased losses of thrust without improving protection of the walls. At the same time excessive thinning of the protective wall layer or its puncture by streams of oxidizer should not be allowed.

In laying out the injectors it is necessary also to provide protection for the head itself from burnout, which might be a result of high thermal conduction from the flame kernel. From this standpoint zones of atomization and evaporation are zones protecting the head

143

from large heat flows, but with great distance between injectors there can be sections of the surface of the head not sufficiently protected from the influence of reverse currents of hot products of combustion, which can lead to burnout of the head. Such danger arises with the application of two-component injectors, for which

the flame front is nearer the head. When swirlers are used the least distance between injectors is usually determined by the size of the injector itself and also by considerations of Strength of the head, weakened by the holes drilled for the injectors, varying from

12 to 30 mm (27). Spray injectors are Given the same spacing.

Flow rates through the single-component injector vary from 30 to 300 g/s and for peripheral injectors can be less. The flow rates through the two-component injector can reach 2.5-3 kg/s. With cooled heads it is possible to considerably increase the flow rate through one injector. Tests of heads cooled by fuel, with flow rate through one injector of the order of 20 kg/s, gave satisfactory results (67).

For protection of the head from burnout it is possible to use different heat-resistant coatings [69].

Fuel is fed to injectors either through special channels (see Fig. 3.24) or by way of formation in the head of different cavities for fuel and oxidizer (see Fig. 3.24a, c, а), ensuring approximately identical pressure drop in all injectors.

Heags with supply of components through channels are structurally more complicated and usually are heavier than those with separate cavities. The lower cavity, usually for structural considerations, is fed the component that cools the combustion chamber,

Mounting of injectors. When injectors are made in the form of

separate structural subassemblies, the most widespread method of mounting 1s soldering. Threaded connection and beading are also used,

144

Ranier,

Placement on a head of various auxiliary devices (igniter, scavenging devices, etc.) is undesirable since they reduce the useful area for placement of injectors and upset the uniformity of fuel Gistribution. In addition the points at which these devices are

mounted are less well protected from heat fluxes, thus creating the danger of head burnout.

3.6. Influence of the Head on Carburetion and Specific Thrust

Investigation of the operation of ZhRD showed that the quality of the carburetion process and specific thrust depend on the design of the head, its shape, and the location and type of injectors. The placement of the injectors likewise influences conditions of burning and heat exchange in the chamber of the ZhRD.

Let us consider certain analytical schemes making it possible to conduct quantitative appraisal of the influence of injector placement on development of processes of mixing of components and on parameters of the ZhRD [8].

Qualitative Picture of the Process of Carburetion in the Chamber of the ZhRD

After entering the combustion chamber the oxidizer and fuel are mixed in both liquid and gaseous states.

One of the basic factors determining the mixing of components is the mutual location of oxidizer and fuel injectors. After the oxidizer and fuel leave the injectors there is collision drops of components and merging of them in addition to partial interpene- tration of atomized streams, promoting mixing of components near the head. Most complete mixing takes place when injectors with crossing streams are used. However, even with parallel axes of injectors, as a result of the meeting of Spray cones at a certain distance from the head, mixing and merging of drops also takes place.

Let us examine, for example, how the mixing of drops of components fed by swirlers occurs.

145

Let us assume that fuel and oxidizer injectors are alternated,

as is shown in Fig. 3.30. With the crossing of spray cones of components at point A part of the components merge 4nto one bundle,

while part of it interpenetrates through the spray cone and continues

-5 move in its former direction. The degree of "penetrability" of

che components depends on the distance between injectors. With increase of the distance between injectors the degree of penetrability is increased. However, for small distances it 1s possible to consider

the sprays impenetrable. Therefore we consider that with the first crossing of spray cones at point A merging of component 0 and into one bundle AB occurs, with very insignificant penetration, leading only to certain expansion of the bundle. The direction of bundle AB is determined by the momenta of oxidizer and fuel.

ore0 гог"> го г

е М ен cliitle

Fig. 3.30. Mixing of drops of components: a) momentum of oxidizer greater than that of fuel; b) momenta of oxidizer and fuel equal.

If the momentum of the oxidizer is greater, the bundle is deflected in the direction of injector Г, and at point В secondary crossing of the bundles of components occurs. Here a spray of the mixture of components, limited rays BC, forms (Fig. 3.30a). With equality of momenta of components the direction of the bundle after crossing at point A remains axial (Fig. 3.30b).

Thus, as a result of the mixing of streams of components emanating from swirlers merging and partial interpenetration of drops takes place. Depending on the momentum of components, the streams cross at one (point A) or two sections of the combustion chamber (points А апа В). In both cases over the cross section of the chamber the "sphere of mutual influence of streams" during the mixing drops

146

$

ыы

Нин

ea

is limited primarily by the spacing between injectors, and at great distances from the axis of the injector the influence of the given injector on mixing is small. Consequently, if the placement of injectors in the head from the very beginning does not ensure

uniform relationship of components over the cross section of the chamber the mixing of G@rops will smooth out this irregularity basically only within limits of the spacing between injectors,

The second factor promoting mixing of components is turbulence, appearing during evaporation of drops.

The intensity of turbulence here 1s influenced by: a) the difference in speeds of drops ard gas;

b) outflow of gases forming during evaporation of a drop normal to its surface;

с) irregularity of intensity flow over the cross section of the chamber;

ad) difference in rates of evaporation of drops of components.

Let us consider the influence of each of these causes on mixing. As was noted above, directly at the head evaporation of Grops starts as a result of reverse currents and radiation (see Fig. 3.4). While the amount of gas forming as a result of evaporation 1s still small drop speed Wy 1s greater than gas Speed wn. With evaporation of drops the speed of the ваз increases and in some section of the chamber the speeds are levelled, 1.e., Wp © We (point M in Fig. 3.6). The quantity of evaporated components here attains 10-35% of the total quantity. With further motion of drops and gas the drop speed lags behind the Gas speed, and accordingly the speed of the gas directly near a drop (or of the gas near formed bundles of drops) also lags behind the speed of the basic flow of 5ав иг. The difference in Speeds promotes turbulent mixing of the gas surrounding the drop (or bundle) with the basic flow of gas.

147

ee нравы АЗЫ

Presence of the stream of gas normal to a drop and forming as

&@ result of evaporation of the drop also promotes turbulent mixing of gas with the basic flow.

Mixing of gas as a result of the shown causes occurs basically |

at distances of the order of the spacing between injectors H. The | influence of this mixing we can consider on the basis of the | corresponding law of distribution of components around the injector | axis.

During mixing of gas nonuniformity of distribution of flow intensity r results in the flow of gases for distances of the order of the chamber radius, since even after the leveling of r at distances from the axis of a bundle equal to the spacing between injectors nonuniformity of r over the cross section of the chamber remains. Here i gas flows to sections, in which flow intensity r is less. However, this process does not render decisive influence on change of distri- bution of components over the cross section of the chamber.

The distinction in speeds of evaporation of drops of component is also evident in turbulent mixing of gases. During the outflow of gases from drops the gases do not entrain liquid drops, and therefore when there is considerable difference in the rates of evaporation | of drops of components (for example, of liquid oxygen and kerosene), j there is a change in distribution of v and r, since the gases of the component evaporated earlier try to distribute themselves more evenly over the cross section. However, due to very high intensity of heat supply to drops in the chamber of ZhRD the rates of evaporation of drops are so high that the difference in time of evaporation : of different components is very insignificant and does not succeed in significantly influencing the distribution of v and r over the | cross section of the chamber as a whole.

Thus one may assume that the mixing of components as a result of evaporation, just as during the mixing drops, occurs basically within the limits of dimensions of the order of the spacing between injectors. Insignificant propagation of mixing of gases during evaporation over

148

“a

течении ртр

larger distances can be allowed for through proper selection of the law of distribution of the component around the injector axis.

After termination of evaporation further mixing and combustion of components occur. However, up to defined chamber lengths full mixing in a chamber, i.e., complete levelling off of the composition of the gas over the cross section of the chamber is not realized.

During further motion of the products of combustion through the nozzle distribution of the relationship of components remains practically unchanged, first, due to the fact that the time of their presence in the nozzle is small, and secondly, because the intensity Qf turbulence in the flow core in the nozzle is considerably less than in the combustion chamber.

Thus if uniform composition of components over the cross section of the chamber were not ensured at the head by the injectors, the turbulent mixing of gases in the combustion chamber and in the nozzle would only smooth the "peaks" of distribution of v and r between injectors (Fig. 3.31), but does not eliminate as a whole irregularity. It is obvious that the greater the length of combustion chamber and nozzle, the less valid this conclusion, since the flow of gases through the chamber and nozzle certain mixing nevertheless continues, but up to a defined length of combustion chamber and nozzle for approximation one may assume that the composition of separate streams of the products of combustion leaving the nozzle is different, where this irregularity is determined in the first place by the lo-ation of injectors on the head.

For appraisal of the distribution of v over the cross section of a chamber it is sufficient to consider the distribution of v obtained as a result of mixing of drops at the head upon exit of components from the injectors. Here the influence of turbulent mixing of gases can be considered by means of the corresponding law of distribution of a Component around the injector axis.

149

Fig. 3.31. Typical distribution of v over the cross section of

a chamber: 1 — neglecting smoothing of peaks; 2 — average values of v.

Distribution of Components Around the Injector Axis

Let us-examine the head with swirlers. For simplicity of analysis we will take checkerboard placement of oxidizer O and fuel I injectors (Fig. 3.32b).

Fig. 3.32. Formation of bundles of mixture of components: a) two injectors; b) checkerboard layout; c) honeycomb layout; d) injectors @ — oxidizer injectors; + — fuel injectors; o ~— points of formation of bundles. |

150

у— semmener ees cas es

We will also consider that momenta of oxidizer and fuel are equal. Consequently, after the crossing of torches at point A the bundle will be axially directed (see Fig. 3.30b). The crossing of torches of two adjacent injectors 0 andTI follows hyperbola A, AA, (Fig. 3.32a). The projection of the line of crossing of torches on the plane parallel to the head will be straight line AA.

After merging of oxidizer and fuel on the line of crossing A, AA, drops have component velocities directed vertically (along the axis) and horizontally. The horizontal component of velocity will obviously be directed from the line connecting the axis of injectors to points А and Ао. As a result at points Ay and A, bundles of the mixture of components are formed.

During the interaction of injectors arranged in checkerboard fashion merging of the mixture of drops into a bundle occurs at points Ay; Ay, Аз» Ay» where in each of these bundles there will be merging of components from the crossing of spray cones of the four injectors surrounding the bundle (Figs. 3.32b and 3.33).

Fig. 3.33. Formation of bundles at head with checkerboard placement

of injectors: e — oxidizer injec- (HOO Efe tors; + — fuel injectors; o — points of formation of bundles.

With honeycomb placement of injectors (see Fig. 3.32c) bundles will be formed at points Ay; Ay, Аз» Ay» Ass and Аб.

In all cases of drops nearer the wall than the axis of the injectors nearest to it strike the wall (see Fig. 3. 32d).

151

Bundles of drops formed in our case at points A)» А, А, and Ay move parallel to the chamber axis. Here the bundles expand both as a result of entrainment of drops by gas and as a result of certain very slight penetration of drops, and also as a result of the evaporation of drops of components.

For approximation we will assune that at sufficient distance from the head the distribution of components around the axis of a bundle follows a certain law of distribution, close to Gaussian law (Fig. 3.34), where the value of root-mean-square deviation of oxidizer or fuel from the axis of the bundle is proportional to the spacing between injectors H, 1.е.,

” a эт "м, (3.83)

where G — flow rate of component in bundle; r — distance from axis of bundle; dG — quantity of cumponent in area dF at distance r from axis of bundle.

Fig. 3.34. Accepted distribu- tion of component around the axis of a bundle.

Let us examine the distribution of a quantity of oxidizer around the axis of a selected injector. The oxidizer fed through the injector was distributed equally between bundles А}, А, Ay, and Ay. Each of these bundles, furthermore, received oxidizer from injectors B,, B., Bs. and B, (see Fig. 3.33). The quantity of oxidizer at the axis of the injector examined by us is determined as the sum of the quantities coming from bundles Ay» Ay,» А, and Ay (Fig. 3.35, curve 1).

152

Fig. 3.35. Distribution of component around injector axis.

nh

Distribution of component around the axis (curve 1) also follows а law close to Gaussian (curve 2). Therefore with sufficient degree of accuracy it is possible to consider thet the distribution of oxidizer around the injector axis follows curve 2, 1.е., obeys Gaussian law, where the root-mean-square deviction of the component 1s picrortional to the spacing between injectors H:

i "м , (3.84)

where Gy — flow rate of component through injector; aGy - flow

rate of component through area dF, normal to the injector axis and at distance r from the axis; k - proportionality factor, determined through integration of equation (3.84):

6, | = паев] dy \ re Та, whence

tne. (3.85)

Taking into account equation (3.85), we express the law of dietri- bution of the component around the injector axis by equation

a (3.86)

If the momentum of drops of one of the components is greater than that of the other, then, as was shown above, at point B bundles of components cross a. second time (see Fig. 3.30).

153

— с Oa Nleematee ts sah umm

Analysis shows that the secondary crossing does not introduce essential change, and expression (3.86) remains valid for appraisal of distribution of the component around the injector axis for both checkerboard and other injector arrangements.

Quantity of Component Reaching an Area of Cross Section from a Single Injector

Let us isolate in the cross section of the chamber an arbitrarily located area and determine the quantity of component Gpy, reaching it from injector 5, located at distance r from the area (Fig. 3.36). From equation (3.86)

Ou 328. (( така, (3.87)

Fig. 3.36. Determination of quantity of component reaching an arbitrarily selected area of the chamber cross section.

Since г s x? + у?,

ыы, f oie hg, ~via Cael ean и . Warr) a(t) (3.88)

Expression (3.88) after reduction twice by Hv can be written in the form чин} чьи of} . (ит) «ит)-| e (т) чи) : (3.89)

154

If we introduce a function of form

= Se (тв (3.90)

then, considering that

f te am coe (3.91)

where z = x/Hv¥2, and designating

wa et tam Se: er oe фа

we obtain: ме 59) ee i Wil 2) eine fell ные ен

(3.93)

Then expression (3.89) can be rewritten:

ФФ де ый © (tn (3.94) Numerical determination of Gry requires the ability to calculate integral (3.90) for any values of 4. Since integral (3.90) in finite

form is not expressed through elementary functions, for calculations we use Table 3.1 of values of (¢) for dirrerent values of $.

155

ед�< hit ae eae

Table 3.1. Values of function 0(z).104

ини

ele | [34| в eur 7 2 9 0.0 | © | 113 | 296 | 306 |4 |554 | ет т | м |103 АТ ТТ 1600 | 1780 | 1900 | 0 | gue 02 DH: 2 29 |9 | мм |8 03 39 | 350 |400 | aie? 04 чит |457 | 5 | ип 05 S716 | 5700 | 50 | 5050 os и #4 |656 | 0655 | 6708 07 ; В | 717% | 7298 |150 | зав 9,0 1 776) 7814 7067 718 0,9 153 ом |1 |354: |в5 1,0 $ 5 | 600 |075 | 9768 1,2 1 0 |900 |948 | 00% 1a о | 9252 | 0075 | 007 | 90 14 7 | ми | 06% | 057 | 0640 16 пе | 97% |975 |9% | мы 16 м | м | 6 |005 | 952 1,7 | 0872 9877 9882 O65 18: 1 | 01s | Oe | com | 00% и 2 | 94 |097 |000 | 0 ‚ $0 н у, 4, 9964,2 9085 ,8 9067,3 oes A РО == „Я вот, | о9тв,5 | 0970,5 | ввоз as cab | c006,: | 9006.7 | ид | 9008.0 2.3 el ‚| 9091,5 | 9000,0 | 9002.4 | 9002.8 2.4 Зри. 9006,0 | 9905.2 | 0005,5 | 9005.7 3.3 „1 :779006,9 9907,1 | 9907.2 | 9007.4 | 9097.5 2.6 ‚0008,2 9008,3 | 9008.3 | 9008.5 | 9008.5 2.7 . ’ 9009, 1 9999.1 9909,2 9000.2 2.8 ‚4000, 4] 9090,5 | 9909,5 | 9900.5 | 9000.5 2.9 , 7] 9000.7 | 9009,7 9900,8 | 9000.8

With change of ¢ from 0 to » function ®(¢) changes from 0 to 1. Substituting in equation (3.93) values of 6, 93 9х1; $ 1; $ , , у, у,2 and determining from Table 3.1 corresponding values of function

@ (4), ме can, using formula (3.94), determine the quantity of component reaching the area from injector Б.

In calculating distances Хх], хо, Yi» Vo it 1s necessary to take them with their sign, and the condition (taking sign into account ) Xo > Хх; У. > У).

156

Relationship of Components in Flow Core

In accordance with formula

and fuel Gn nn? reaching an arbitrari)

Section area of the chamber section will be equal

(3.94) quantities of oxidizer 65 пл y selected area in the cross

from 1p and in of the injectors

==: 6%. [6(1)— © (9.1) (Ф (%.2)— $, 11; Cram + 7 б., [(9,,2)— Os, )][@ (%.2)— (9,,1)}. | {3.95)

The mean value of the relationshi

р Of components passing through the area

о Ж.д, Ven 9% OL oe

Onas (9.1) — 9 (es TF (2) 908, (3.96)

If flow rates through all injectors 0 апа Г are identical,

% 9, Sane? Gynt

(3.97) and expression (3.96) will take form г $ [9.4 —© (9..1) © (%,2) — (9,1) ; Yan Vey le т). и. (3.98)

Р Formula (3.98) gives us the avera

of components passing through the arbi chamber,

ge value of the relationship

trarily selected area of the Here the dimensions of the area were not limited,

For practical calculations it is

fully sufficient to take linear dimensions of the area e

Relationship of Components Уст in the Near-“all Layer

As noted above,

all drops of components exceeding the limits of the line connectin

@ the axes of the extreme injectors strike the

157

Bal: умы

отв тонах

БЫ авы

аль.

wall. Since in striking the wall all drops remain within the limits of the area, one may assume that the quantity of component reaching а near-wall area of width Xo - X) (Fig. 3.37) will be equal to the quantity of component in a strip of infinite length (1.е., Yo * „).

Fig. 3.37. Determination of Yon

When Yo = * function Po, 2) = 1 and the expression of (3.95) 3 for the area located at the wall will be recorded in the form:

Gane Gof (92.2) — 9 (2,1))[1 — © (ey) .0)};

’ (3.99) Cee A - бы [1 —Ф (9). г

Hence the ratio of components in the near-wall layer

. “ 2% [* (9х1) —@ (91,1) (! _® (%.1)] то ты [® (9+.)—Ф (9-9, ен

Just as during determination of v for average sections of the chamber,

during determination of vo, the size of the area (x, - 8,) is

expediently taken equal to the spacing between injectors. Formulas : (3.95), (3.96), and (3.99) and (3.100) permit determining flow

rates and the relationship of components with account be taken of

all injectors, in a head. However, as one may see from Fig. 3.35

the influence of injectors apart from the area by a distance greater

than three spacings is very insignificant. Therefore during calcula-

tion of the shown parameters it is fully permissible to consider only

158

vi

injectors distant from a given area (in middle of the cross section or at the wall) by no further than three spacings.

Usually arrangements of injectors on a head have axial Symmetry. Therefore there is no need to calculate flow rate and the relationship of components for all sections of a chamber, but it is sufficient to calculate only for one sector of the head (for example, with checkerboard layout for 1/8 of the cross section). Since the influence of a large number of injectors located at various distances is considered here, for practical purposes it is convenient to perform calculations by using a scale diagram of the injector layout and measuring distances Xo» Xp» Vos Vy directly on the diagram.

Example of calculation of relationship of components Yor Find the relationship of components at the wall in section No. 1 of the combustion chamber head of an engine. A plan view of

injector placement and the location of the selected section are shown in Fig. 3.38.

‘Fig. 3.38. Solution of example: o - screen injectors [,; + — fuel injectors

Г; о - oxidizer injectors 0.

159

Flow rates through the injectors: screen fuel Gni3 2 10 g/s; fuel Gn = 33 g/s; oxidizer 6, = 90 g/s.

Solution: We consider that the section is reached only by components from injectors located at distance <3H. Drawing an arc with radius equal to 3H, we define the region containing injectors influencing the relationship of components passing through area No. 1. We see that this region contains 13 injectors.

For convenience of further calculation of data on flow rate, distances, and values of auxiliary functions for these injectors is given in Table 3.2.

Having determined all necessary data, we calculate the relation- ship of components in section No. 1 (3.100):

[9 (9:4) — © (9,1) [1 -— © (ти) Е [9 (9х) — © (9л1)] [1 — © (9,1)

7,08 + 2-2,60 + 2-0, 116 + 9-1,26 + 8,034 2-0,733 “ °

a

It is possible to determine the relationship of components in any other section analogously.

Geometric Method of Calculation of Relationship of Components Over the Cross Section of a Chamber

Calculation of the distribution of the relationship of components over the cross section of a chamber is quite labor-consuming. Therefore for tentative calculation of the distribution of v andr over this section it is possible to use simplified geometric method, the essence of which consists in the following. The entire area of the head is split into sections. As limits of the sections are usually taken lines connecting the centers of oxidizer injectors . (Fig. 3.39). For each of the sections we determine the number of included oxidizer 4, and fuel 1, injectors. If part of the injectors are on the boundary of sections, the i, or in can be and fractional, since in this case the separated section receives only part of a component from a given injector.

a

160

aon em

23 # : i : i ails wa =z = H м

= i 1 CHDe | (De 6") о | СФ a i = = мет ги ss г =

FE = = : : ^^ == (5) Фх № сага $ зо ова�'еА ресшузтр ов�<

*2�&�& этавь

а

161

а

"<

м“

S

Fig. 3.39. Determination of local values of v by geometric method.

Relationship of components v and flow intensity r are determined by formulas:

4G, 2 (3.101) 2 2140. + уго, ‘ Py (3.102)

where 6, апа в. — flow rates through one injector of oxidizer of fuel respectively; Fyy — area of section. For an example let us examine sections 010,030, (Fig. 3.39a) and 0103030,0,0¢ (Fig. 3.39b).

Section 0102059, from each of the oxidizer injectors 45 fed only 1/4 of the flow, while section 01020300505 receives 1/3, For the shown sections the number of oxidizer injectors 4, will be 1/4 х Ц = 1 апа 1/3 x 6 = 2 respectively. The number of fuel

injectors 1, = 1 for both sections,

The relationship of Components the пеаг-иа11 layer is calculated

analogously (see Fig. 3.39b). For section Aj A,

kB 20 \= 3+3 +3 Ft Ts bod vee

For section АА.

в в. а 8. ow STS ST tT tpg: bes oo In this method no allowance is made for the influence of injectors located ‘beyond the limits of the section, although, as was noted

~ above (see Fig. 3.35), such influence prevails and most significantly

affects the relationship of components in the near-wall layer,

162

———

Therefore the

Components gives less reliable results and for calculation of the relationship of components at the wall is entirely unsuitable.

Influence of Irregularity of v and r Over the Cross Section of a Chamber on Specific Thrust and Complex 8

Thus it has been established that in any of the sections of the chamber relationships of components v and flow intensity r are distributed nonuniformly. In other words, in certain sections the values of v and Yr differ from calculated average values. Deviation in some section of values of y from the average (most frequently optimum) means that in these sections true combustion

temperatures and the composition of combustion products also differ from those calculated.

If we assume that the selected of components over the entire chambe to the biggest value of Руд» specifi values of v differ from optimum in e and the specific thrust of the combu Obviously, will also decrease.

mean value of the relationship

ris optimum, 1.е., corresponding c thrust in sections where

ither direction will be lower, Stion chamber on the whole,

the values of flow rates, relationships of components, and flow intensity for each of the sections, we find the specific thrust of the engine

and the computed value of complex 8

as the average value of specific thrust and

complex ay of separate streams of different composition:

. ый ya (3. 103)

мо „- 2% (3.104)

where Руда — Specific thrust for the

true relationship of components in the i-th section; G

4 ~ flow rate of fuel through the i-th section;

163

ene ps aes

@ - total fuel flow rate; By — value of complex 6, corresponding to composition of components passing through i-th section.

164

CHAPTER IV COOLING OF ZhRD 4.1. Heat Exchange in ZhRD

Organization of chamber cooling is one of the most importarit problems in designing the ZhRD, and as compared to other types of

heat engines is complicated considerably by peculiarities of the process of heat exchange in such engines.

The first peculiarity consists in that the process in the chamber of the liquid fuel rocket engine takes place at high temperatures (3000-4000°K and pressures (to 100 [atm(abs.)] and more). Since combustion products move through the engine chamber at very high speed, the coefficient of convective heat transfer from hot combustion products to chamber walls and convective heat fluxes dx» reaching in the critical section of the nozzle 20-60-10° keal/m°h (23.3-69.7-10° w/m?), increase sharply.

The second peculiarity of heat exchange in the ZhRD is the

high level of radiation in the combustion chamber, which leads to large radiant heat fluxes Чл..

As it is known, the emittance of gases is proportional to the absolute temperature in degree of 3-3.5, and therefore at the shown temperatures in combustion chamber and nozzle of the ZhRD large _ heat fluxes are caused by radiation Чл» which for conventional propellants reach 20-40% of the total heat flux to combustion chamber walls. With reduction of the temperature of gas in the

165

АИ

nozzle the relative part of radiant heat flux decreases.

The third peculiarity of heat exchange consists in that powerful total convective and radiant heat fluxes to the chamber wall cause its temperature to reach impermissibly high values. Therefore in the ZhRD materials possessing heat resistance and greatest possible thermal conductivity should be used — but this, however, is very

difficult since heat-resistant materials, as a rule, have comparatively low thermal conductivity.

The fourth peculiarity of heat exchange is due to the conditions of application of the ZhRD аз а propulsion system for flight vehicles (rocket, satellite, and aircraft). Therefore it is not rational to use special liquid for cooling such engines. Usually the ZhRD is cooled by one of the components of fuel by passing it prior to its entry into the combustion chamber through the cooling cavity. Such principle of cooling complicates chamber design and imposes additional requirements on fuel components, since the quantity of the component passed through the coolant passage is limited by its flow rate,

In addition to the indicates chief characteristics of the ZhRD on the whole the process of cooling is also influenced by the kind of fuel, type and assignment of engine, and its design.

Thus the application of a low~boiling pair of components (for example, oxygen + hydrogen) means that the chamber is cooled basically not by liquid, but 5а8, resulting in additional requirements being imposed on the engine design. The application of nozzles with central body sharply complicates the problem of cooling the ZhRD in connection with increase of the surface of the combustion chamber, and especially the perimeter of the critical section, in which heat flux is greatest.

A number of specific problems in the organization of cooling of ZhRD arise during the use of different types of fuels, applica- tion of different chamber designs and coolant passages, and during the use of different forms of cooling.

166

ae

These problems will be discussed below in greater detail in corresponding sections.

The temperature of the chamber walls of the ZhRD can be held within permissible limits by one of the following methods:

1

External (or regenerative) cooling. é. Internal cooling. -

3. Mixed cooling.

= °

Radiation cooling.

5. Ablation cooling.

Shielding internal walls with heat-resistant coverings.

nN

~

Protecting walls from burnout by accumulation of heat. Heat Exchange in the ZhRD with External Cooling

In order to understand processes of heat exchange in the ZhRD we will examine the basic factors affecting the effectiveness of cooling in reference to the most widespread method — external cooling.

A typical diagram of development of the Process of heat exchange in the ZhRD with external cooling is presented in Fig. 4.1. Here То — temperature of combustion products, Тг.ст — temperature of chamber wall on the hot-gas side ("gas" wall), Ty og — temperature

of chamber wall on the coolant side ("liquia" wall), Ty ~ temperature of coolant.

Heat is transferred by convection and radiation from hot combustion products 1 to combustion-chamber wall 3. Thus it is possible to say that the total specific heat flux from hot gases to chamber walls Ч; 1s composed of two Specific heat fluxes:

167

WAN et a

convective Vx and radiant Чл» 1.е., 4:=4,-+4» (4.1)

where

GG, (T3—T7, 09). (4.2)

q Fig. 4.1. Diagram of the process ‘eer of heat exchange in the chamber of ZhRD. 3 7 =. ны

Thanks to thermal conduction heat is transmitted through the chamber wall and then by convection to coolant 2, passing through the cooling cavity.

At the initial moment of operation of an engine, when chamber and nozzle walls are still cold, part of the heat flux goes to heating the chamber walls. Such are the cooling conditions of the ZhRD for which wall temperature and heat flux into the liquid coolant change with time are called nonstationary cooling conditions.

As time passes equilibrium is established. Here the entire heat flux from hot combustion products to chamber walls is removed. From this moment (under given engine operating conditions) the temperature of the gas and liquid walls of the chamber and heat flux through the walls remain constant. Such conditions are called

steady-state cooling conditions.

We will subsequently examine only steady-state cooling conditions.

168

Dependence of Ther on Coolant Speea

The speed of liquid coolant w affects the value of the

coefficient of heat transfer from wall to 1t ay, where with increase of this speed ay also increases.

With the same specific heat flux Gd; and temperature of liquid

Ty the temperature of the wall on the liquid side Tact is determined by expression

= (Ти Гы), (4 : 3)

which after conversion takes the form

Треть, (4.4) 1.е., with increase of Gy temperature Ty op decreases. In turn

heat flux is transmitted through the chamber wall by thermal conduc- tion, 1.е.,

A

where y — coefficient of thermal conductivity of the chamber wall material, whence

Теа ae (4.6)

Thus for the same metal wall thickness 6... and specific heat i

flux Ч; the temperature of the gas wall Ther will be lower, the з lower the temperature of the wall on the liquid side Тж. ст: 5 . 3

x

However, with decrease of Ти. ст total heat flux from gases to chamber wall will increase somewhat as а result of the decrease of Ts oe (4.2), which leads again to growth Th om Tr om ete.

Finally, steady-state cooling conditions after increase of р speed of the liquid coolant are established at smaller Tp cr and

169

= Е — 2

"Ж.ст than prior to the increase of speed (Fig. 4.2a). Thus the temperature of the gas wall Troer in great degree depends on the speed of the liquid coolant: the greater the speed, the less is Thor Consequently, necessary Тр.ст can be maintained by increasing the speed of the liquid coolant w, for example, by decreasing the cross section of the coolant passage.

SS

Fig. 4.2. Dependences of Tp,o7:

a) on speed of liquid coolant; b) on coefficient of thermal conduc- tivity of metal 1; ---- at high values of w or i.

+ |

КИДАЛА

SAS

Influence of Surface Boiling of Liquid Coolant on the Value of Trocr

In the process of heat transfer from the wall of the combustion chamber to the liquid coolant two cases of the relationship between Teor and the boiling point of the liquid т at a given pressure in the cooling cavity are possible:

Тис «Та Ти. > Т.. (4.7)

In the first case ebullition of the liquid in the coolant passage is impossible. In the second case the liquid coolant can boil on the surface of the liquid wall of the chamber. If the mass of liquid here has temperature below Тз› bubbles of vapor formed on the surface of the liquid wall bubbles of vapor will be washed off by the stream of fluid and condensed in the colder flow core. Thus these bubbles, agitating the stream, will increase the heat transfer from wall to liquid, 1.€., ay will increase for the same speed of the cooling liquid.

170

The increase of Gm due to surface boiling will lead to the same results as an increase of Gx due to an increase of speed,

1.e., to a decrease of Tr.or With a certain increase of total flux aq;

Conditions of cooling of the wall in this case can be improved without an increase of the feed pressure of the liquid coolant for increasing the sp2ed of 1ts motion. However, intense boiling on the surface can lead to such vigorous vaporization that the bubbles of vapor, not being washed off by the fluid, will form a solid film of vapor on the surface of the wall. Since vapor conducts and transmits heat poorly, in this case the coefficient of heat transfer ay decreases sharply, leading to the reverse result — to increase of Theor and, possibly, to burnout of the wall.

Dependence of Tn op on the Thermal Conductivity \ of the Material of the Chamber

As can be seen from relationship (4.6)

T,. ст 03 “e + Tuer

for the same values of Тж.ст» 6 ст апа 9; the temperature of the gas wall T non Will be lower, the greater the coefficient of thermal conductivity of the wall material .. However, even here with decrease of Tp cr in accordance with expression (4.2) total heat

flux 9; will increase, as a result of which Tr.cris again raised Somewhat. In this case (just as in the case of lowering of Ty on with increase of a») steady-state cooling is established at a smaller

value of Tp or, than would be the case for the wall with smaller value of i.

As follows from expression (4.5), the curve of temperature drop through the wall of an engine with growth of A will be more sloping. Since q, increases somewhat, Ty o7 will also increase (4.4).

171

мании ар ле м

a

Thus the line of change of temperature through the wall for metal with greater \ intersects the line of change of temperature through the wall corresponding to smaller \ (see Fig. 4.2b), Consequently, for the manufacture of chambers of the ZhRD it is expedient to use materials with greatest possible coefficient of thermal conductivity i. However, as a rule, such metals rapidly lose their Strength with temperature increase.

Dependence of Ther from Thickness of Wall of Engine Chamber Son

From expression (4.6) it also is clear that with constant

Ти. ст and 9; the temperature of the gas wall Throop Will be lower the less the thickness of the wall бот.

Although with decrease of Trop heat flux dy» according to expression (4.2), increases Somewhat, steady-state cooling conditions are established with smaller Thor Consequently, the chambers of ZhRD should be made with smallest possible wall thickness бот.

The Influence of Pressure and Temperature in the Combustion Chamber on Values of Troer and Ч;

With increase of pressure in the chamber the density of the Gas moving in it is increased. In connection with this the coefficient of heat transfer from gas to wall a, and the value of convective flux dn (see section 4.6) increase.

А8 increase of pressure in an engine chamber will also lead to growth of partial pressures of components Pro and Poo the 2 2

radiation of which makes up the radiant heat flux to the wall Чл (see section 4.8),

Thus an increase of pressure leads to an increase in total specific thermal flux tothe chamber will q and, in accordance

with formulas (4.4) and (4.6), to an increase of Ty op and Тр. ст.

172

It is true that with growth of Th cr heat flux d; Somewhat

decreases, but a new steady state is established at new, greater values of Тр с and 9; .

T Thus increasing the pressure in the chamber of an engine, other things being equal, leads to an increase of Theor and Qs:

With an increase of Ts convective and radiant heat fluxes

increase, which, as does an increase of pressure in the chamber, leads to growth of Ther and Qs:

Therefore frequently for ZhRD working on fuels with greater calorific power Hy, more intense chamber cooling is also required.

Influence of Dimensions and Shape of Coolant Passage on Tr on

With decrease in the cross section of the coolant passage coolant speed is increased.

The influence of the shape of the coolant passage is due to the ribbing effect. The presence of longitudinal or helical ribs, corrugations, tubes, etc., increases the surface of heat removed from the coolant side. The latter leads to increase of the effective coefficient of heat transfer from the liquid wall to the coolant, and consequently to a decrease of Тг.ст. The degree of influence of the ribbing effect on Treen in great measure depends

on the thermal conductivity of the material of the inner shell of the engine chamber (see section 4.11).

Influence of Engine Operating Conditions on Тр. ст

During operation of ZhRD under different conditions of chambers

cooled by one of the components rated conditions are usually those of greatest thrust. —

During transition of an engine to operation at lower thrust the pressure in the combustion chamber Po drops and �>> decreases

173

a виа,

— ants a

somewhat, as a result of which heat flux 9; decreases. However,

with decrease of thrust there 1s a simultaneous decrease in the flow rate of the coolant component, and consequently of the speed of the coolant through the passage w and of the coefficient of heat transfer from wall to liquid ay. Since the cooling surface of the chamber remains constant, the decrease in flow rate of the coolant can lead to such growth of Ty, that the coolant can boil.

Thus with reduction of thrust, in spite of the decrease of heat flux Ч; › the temperature of the gas wall Tp oy can increase. Therefore, if an engine must operate with variable thrust, it is necessary to check out cooling under conditions of low thrust.

Having examined the basic factors influencing the effectiveness of external chamber cooling, one can make the following conclusion.

For decrease of Tp oq with external cooling one should increase the speed of the liquid coolent. It is desirable that the chamber be made of metals with greatest possible thermal conductivity and with least wall thickness and maximum cooling surface. Cooling must be effective under engine operating conditions.

If it is impossible to ensure permissible value of Tp op by external cooling alone, it is necessary to decrease heat fluxes to the wall, or to employ additional other forms of cooling (internal) or to protect the wall with heat-resistant coverings.

4.2. Methods of Cooling the ZhRD External Cooling

The simplest arrangement for external cooling by one of the components of fuel is shown in Fig. 4,3a. The liquid coolant enters collector 1 and passes from it into the coolant passage. In flowing through the passage the liquid cools the walls and is itself heated. The heated liquid leaves the cooling jacket and enters head 4. As was already pointed out, with external cooling the chamber of the

174

So RNIN Oa oe seomes oe RRM OTT mee eens on

engine is usually cooled by the components themselves. The liquid coolant can be fuel or oxidizer.

Qridizer

Fig. 4.3. External cooling arrangements: a) by one component; b, c) by two components; 1 — inlet collector: 2 — coolant passage; 3 — outlet collector; 4 — head.

It is necessary to note that for every fuel and chamber pressure Po there corresponds a minimum thrust, below which external cooling alone becomes insufficient. The cause of this is the fact that with decrease of nominal thrust (with constant Po) the necessary volume of the combustion chamber decreases in proportion to the total consumption of fuel, while the cooled surface of the chamber decreases only in proportion to the 2/3 power. Thus the lower the nominal thrust of an engine, the greater the ratio of the cooled surface of the chamber to its volume. Furthermore, with decrease of critical throat diameter Dp» other things being equai, normal to the wall convective heat flux qy, according to formula (4.181) — All of this leads to the fact that in chambers of low thrust the coolant receives a greater part of the heat developed during combustion of fuel than in high- thrust chambers. Thus for a combustion chamber with a thrust of the order of 500 kg (*5000 N) and working on hydrocarbon fuels paired with oxygen or nitric acid for p, = 20 КЕ/ст? (%1.98 MN/m° ) the quantity of heat going into the coolant is approximately 2% of the

increases in proportion to D

175

aera: LENE NE

total quantity of heat generated. For the engine of the A-4 missile, having a thrust of 25 T (40.25 MN), this heat amounts of only 0.7%, 1.е., the greater the thrust of an engine, the more easily external chamter cooling is accomplished.

For comparable thrust and pressures the higher the rate of full conversion of fuel into products of combustion, the smaller the volume necessary for combustion of the same quantity of fuel, and Consequently the smaller the Surface that must be cooled. Hence the lower limit of minimum thrust, for which external cooling is effective is lower, the shorter the conversion time.

For normally used fuels this limit is considerably below 500 kg (%5000 N), while for slow-heating fuels, requiring more time, external cooling can be insufficient even with high thrust.

are sometimes used for cooling. Here one of them cools the nozzle of the engine while the second cools the combustion chamber (Fig. 4. 3b). Sometimes the second component is used for lowering the temperature of the component directly cooling the chamber of the engine and is itself heated (Fig. 4.3¢), External cooling by components also can be applied during the use as fuel of vapors

of low-boiling components (for example, oxygen + hydrogen). The Specific character of such external cooling consists in that the coolant is in liquid state only in the initial section of the

coolant passage, and the main part of the chamber Surface is cooled by gaseous coolant.

In Fig. 8.7 is given an example of a simplified arrangement for cooling a chamber with a low=-boiling component — hydrogen. Liquid

hydrogen enters pump 6 and passes from there into the coolant passage, where it is evaporated, so that the greater part of the chamber

surface is cooled by gaseous hydrogen, which is heated in the process to 200-250°K (see section 4.13).

Furthermore, there have been projects proposing the cooling of ZhRD of high thrusts by water in such a way that the water in the

176

Coolant passage is heated to the boiling point [25]. The steam formed here enters turbine and 1s used to drive fuel pumps.

Internal Cooling

With internal cooling wall temperature T, 47 is reduced thanks to its protection by liquid film or gas layer of temperature lowered

&S compared to the core and created on the internal side of the wall.

Such layer usually is called the пеаг-иа11 layer.

If for external cooling the problem is reduced to finding how it 1s possible to more effectively remove the heat entering the wall, not protected from the influence of gases heated to high temperature, then for internal cooling by way of creation of near-

wall layer of reduced temperature heat flux into chamber walls is reduced. ‘

The temperature of the gas in the near-wall layer is reduced because of the artificial enriching of this layer by one of the components, which in this case is also the coolant. Usually such component is fuel (although the protective layer could also be a near-wall layer with a Surplus of oxidizer). For internal cooling it is possible to use water or gas also.

The surplus of fuel for creation of the near-wall layer is fed either through injectors, located about the periphery of the head, or through special belts, which can be located both directly at the head of the chamber and in the cylindrical and nozzle parts of the chambers.

With supply of surplus fuel through peripheral injectors (Fig. 4.4a) mixing and combustion of components take place at the wall when a << 1, with formation of a near-wall layer of combustion products which protects the wall from high heat flux. The advantage of such arrangement of internal cooling is implicity of creation

177

мя

of the protective near-wall layer without complication chamber design.

Fig. 4.4. Creation of the near-wall layer for internal cooling: a) with the help of peripheral injectors;

b) with help of cooling belts; 1 — oxidizer; 2 — fuel; 3 — chamber wall; 4 — boundary layer; 5 — near-wall layer; 6 — peripheral injectors;

7 — liquid coolant; 8 — bents of coolant supply.

However, with increase of chamber length the economy of this method of cooling worsens, since the farther the section of the wall 13 from the head, the greater the erosion of the near-wall layer.

For certain engines it is more economical, but structurally more difficult to apply internal cooling by passing coolant through special belts (Fig. 4.4b).

When the coolant is fed through the belt, the mechanism of protection of the wall from burnout can be represented as follows (Fig. 4.5). Under the influence of high heat flux the liquid 1s evaporated, and above the layer of liquid is created a protective layer of vapors of liquid and combustion products (if combustion takes place in the layer).

Thus, in essence, are obtained two protective layers: liquid and vapor. The entering liquid, entrained at high speed by the stream of combustion products, spreads over the chamber wall in a very thin layer.

178

° Pen. г % “0 a % ~ 2е. 7. т, ls. <" ад =

х

СМ

a)

а РР: 1 hale Be ЕН КОСССОВССЕ ЛОР О РОС О РРР

Fig. 4.5. Diagram of the protective action of internal cooling: a) supply through cooling belt; b) sweat cool- ing; 1 — coolant; 2 ~ porous wall;

3 — liquid layer; 4 — vaporous protec- tive layer; 5 — stream of combustion products.

Owing to evaporation, the thickness of the vapor layer above the liquid grows in the direction of motion of the stream of combustion products up to the section where the liquid is completely evaporated. Following this section there is а gradual decrease in the thickness of the vapor layer as a result of its mixing with the basic flow, i1.e., erosion of the Screen, but the intensity of erosion of the vapor layer is relatively low, and therefore the protective action of the layer is preserved over relatively greater distances.

Since the density of the coolant vapor is many times less than that of the liquid, the thickness of the vapor layer also is greater than that of the layer of liquid. Furthermore, inasmuch as the thermal conductivity of the vapor is many times less than that of the liquid, the thermal resistance of the vapor layer is several times higher than that of the liquid layer.

As a result of the Joint influence of these two factors the

protective layer can be considered the vapor layer (or the mixture of vapor and products of combustion).

179

In Fig. 4.6 are given data from experiments on change of heat flux in different sections along the length of the chamber of a ZhRD as a function of the per-second flow rate of the water used on internal cooling.

qa" Fuels 756 + me 5 beal/a? on Coslant = wae т

9 у

Fig. 4.6. Influence of internal cooling on the value of heat flux.

Thus, for example, in section No. 8 (critical) with a flow rate of water of up to 1 g/s heat transfer to the chamber wall remains practically unchanged, since all steam formed here is washed off by the gas stream. With further increase in the flow rate of cooling water a steam screen is formed and heat fluxes to the chamber wall decrease noticeably in direct proportion to the coolant expended on the Screen. However, for coolant flow rates higher than 18 g/s there is no further noticeable decrease of heat fluxes. The cause of this phenomenon is that as s00n as the liquid film reaches the examined section and, thus, the entire surface�' of the nozzle from the coolant supply to belt section 8 1s covered by the liquid film, further increase of coolant flow rate does not change the surface of the liquid from which the formation of the vapor protecting the given section of the wall occurs or the thickness of the protective

180

a

vapor layer above the examined section. This leads only to increase in the thickness of the liquid layer, which, as was mentioned, has no significant influence on heat exchange. Thus the basic problem

during creation of internal cooling 1s creation of a stable screen of steam at the wall.

A variety of internal cooling 1s so-called sweat cooling. In this case the chamber wall is made of porous material and the coolant passes through the pores evenly over the entire surface of the chamber

(see Fig. 4.5). Both liquid and gas can be used as coolant with the porous wall.

With porous internal cooling for creation of a reliable screen an insignificant Quantity of coolant is required. There is no need to make the entire chamber of porous material since a stable Screen of steam can be created by using a porous cooling belt.

The deficiency of available porous materials is that during engine operation the pores can be rapidly obstructed. Therefore it is very difficult to create porous material with time-constant flow friction over its entire surface,

In certain designs of ZhRD internal cooling is carried out either by means of peripheral injectors in the engine head or by means of special cooling belts made in the form of a annular slot or row of holes in the combustion chamber (Fig. 4.7).

Cool unt

b)

Fig. 4.7. Diagrams of belts for internal cooling: a) belt of holes; b) slot belt.

Apparently, it is possible to so organize internal cooling that it will ensure operation of the engine chamber for the necessary time without burnout. However, with internal cooling alone it is

”

181

ee es й

necessary to use such a quantity of coolant that part of it remains unburned, and losses of thrust are comparatively high (see section 3.6). Therefore the combination of internal and external cooling,

mixed cooling, utilized in the overwhelming majority of ZhRD 1s the most expedient.

Mixed cooling makes it possible with a use for internal cooling 1-3% of the total fuel consumption to realize reliable protection of chamber walls from burnout with comparatively small losses of thrust. Here internal cooling is accomplished by means of both peripheral injectors and by installation of cooling belts.

In Fig. 4.8a is a diagram showing the arrangement used to cool the RZ-2 engine, while Fig. 4,8b shows that used in the engine of the A-4 missile. In the first case the near-wall layer is formed by way of corresponding placement of injectors, and in the second four cooling belts are employed.

Fig. 4.8. Examples of mixed cooling arrangements: a) internal cooling by means of cooling belts;

1 — coolant; 2 — near-wall layer; 3 - fuel; 4 - cooling belt.

Other combinations of all the above-described methods of external and internal cooling are also possible.

Radiation Cooling

In engines having large values of Р2/Рз the nozzle there is a sharp decrease in density and temperature of the products of combustion, which in turn leads to considerable decrease of

182

convective and radiant heat fluxes. Here wall cooling can be

provided by means of heat removal from the wall by radiation into the surrounding medium (or vacuum).

Such cooling is called radiation cooling. With radiation cooling wall temperature T,, can be much higher than the thermodynamic temperature of the stream of combustion products. With growth of Top radiant heat fluxes from the wall increase as a result both of growth of Top Of increase in the degree of blackness of the wall Con: AS @ result of strongly increasing radiation of the wall into the surrounding medium (or vacuum) intense cooling of the wall will also take place. Furthermore radiation from the wall into the products of combustion can be higher than radiation of these products and the direction of Чл will be from the wall to the gases which also leads to certain cooling of the wall.

During radiation cooling owing to equilibrium between supply and removal of heat from the wall certain equilibrium wall temperature Tor.p is estimated. If values of Тст.р do not exceed those permissible for the given material, pure radiation cooling of the wall is possible. With values of Тст.р larger than permissible additional protection of the wall by means of internal cooling or

heat resistant coverings is necessary (for details about radiation cooling see section 4.13).

Ablation Cooling

Ablation consists of the processes of evaporation (or sublimation), fusion, combustion, and destruction of the surface

layer of a body, accompanied removal of material by a high-speed gas stream.

In ZhRD wide use has been made of ablation cooling, by which

protection of walls from burnout is provided by putting ablating coverings on the internal surface.

183

a SS a ee

————

a

Since during ablation cooling there is ablation of the protective covering, such cooling is sometimes called cooling by removal of mass. Ablation cooling is used very effectively and widely for burnout protection of the chambers of upper-stage engines, and nozzle caps.

However, for prolonged operation of a ZhRD the weight of the heat shielding material is rather considerable, which is a serious drawback to this method. The second deficiency of ablation cooling 4s change of shape of chamber and nozzle due to ablation, and sometimes due to exfoliation and breaking off of material.

Other Methods of Protecting Chamber Walls from Burnout

A very effective method of protecting the chamber walls of ZhRD from burnout is to heat-resistant coverings on the fire side. K. E. Tsiolkovskiy suggested that the inner walls of these chambers be covered with graphite, tungsten, or other heat-resistant materials.

In Fig. 4.9 is a diagram of the ORM-9 combustion chamber, developed in 1930, in which was used ceramic thermal insulation based on zirconium and magnesium oxides. In the nozzle part of the chamber was a copper insert for accumulation of heat.

Fig. 4.9. Combustion chamber of the ORM-9 engine. 1 — ceramic covering; 2 —- copper insert for accumulation of heat.

The basic deficiency of ceramic coatings consists in the fact that ceramics crack under the influence of high-temperature flux.

184

For the chambers of the ZhRD of the long-duration (of the order of several minutes) it is very possible to use a combination of Close-grained heat-resistant coverings (0.1-0.4 mm) and flow-through external cooling — combined cooling. With combined cooling it is possible to manage without internal cooling and its related losses of thrust, but it 18 difficult to Satisfy the stringent requirements on the covering imposed by conditions of operation of ZhRD. The | covering must have a high melting point, resistance to thermal shocks, low thermal conductivity, a coefficient of linear expansion close to that of the base material, good adhesion to the chamber material, and Satisfactory anticorrosive properties.

= a ™

In Table 4.1 are given the thermopnysical properties of certain refractory materials.

Combined cooling frequently is created naturally in existing ZhRD as a result of deposition of soot on the inner walls of the chambers. It has been found that chamber cooling 1s improved here.

Protection of chamber walls from burnout for a certain time can also be provided by applying materials with high thermal conductivity. In this case the heat entering the chamber walls, thanks to their good thermal conductivity, spreads rapidly through the entire mass of material, is absorbed as a result of heat capacity, and thus is essentially accumulated in the walls. Such method is therefore called protection by accumulation of heat.

The use of accumulation of heat is expedient under such engine operating conditions when short-term operation is followed by a prolonged period of cooling.

To ensure efficiency of an uncooled chamber for a prolonged time (60-100 s) a combination of heat-resistant coverings and heat- accumulating materials ir frequently used. In Fig. 4.10 is given an

example of the construction of the wall of the nozzle part of an uncooled engine chamber.

185

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тешлои лэрип) зТтетлэзеш Алозэелуеа JO зэтзлэао3а теэтзАчцаошалаэцу,

* (=чотз>�рио> “T° эта

186

Fig. 4.10. Will of uncooled nozzle: 1 — tungsten; 2 — carbide layer; 3 — graphite; 4 — ceramic; 5 —- plastic;

6 ~ metal frame; 7 - sheath of plastic filaments.

The internal (fire) wall 1 is made of a thin layer of tungsten, withstanding high temperatures (to 2700-3300°K), and fulfilling only the function of heat resistant protective covering. The graphite layer 3 1s the basic accumulating layer. Intermediate carbide layer 2 serves for prevention of diffusion of carbon into the tungsten shell and accumulation of gas between the heat-resistant wall and the heat-absorbing layer. Ceramic layer 4 and plastic layer 5 are heat-insulating layers for protection of the metal load-bearing frame 6. On the outside of the wall is an additional Sheath of plastic filaments 7. For the uncooled sections of chamber or nozzle the possibility is also being investigated of using porous materials with a filler of fusible materials. Under the influence of high heat flux the filler is evaporated and sweats from the pores, result in formation at the wall of a protective cooling screen.

4,3, Process of Convection Heat Transfer from e Gas to e Wa

Dynamic and Thermal Boundary Layers With the motion of a compressible liquid with high speed along the wall of a combustion chamber and nozzle of a ZhRD, on the surface

of the wall dynamic and thermal boundary layers will be formed (Fig. 4.11).

, 187

Fig. 4.11. Boundary layers: 1 — dynamic;

In the dynamic boundary layer under the action of forces of viscosity the speed of motion of the flow decreases from the speed in the nucleus of flow down to zero at the wall, where it is as though the gas "adheres" to it. In the thermal boundary leyer the temperature of the gas is changed from values in the nucleus of the flow to values at the wall Troon: Depending upon whether there is heat removal from the surface of the wal) or not (heat-insulated wall), the nature of the change in temperature is distinguished, With a completely heat-insulates wall (sometimes it is called impenetrable) quantity Tp ep in an ideal case will be equal to the stagnation temperature of the nucleus of the flow Tp, (Fig. 4.12). With heat removal from the wall Tp 97 will be lower than T),, which is determined by the intensity of heat transfer from the surface inside the wall.

Fig. 4.12. Change in thermo- dynamic temperature То and

Stagnation ternperature Too

over the thickness of the thermal boundary layer:

----- for a heat-insulated wall; with heat removal from the wall.

In general the thickness of the thermal boundary layer A can be greater than, smaller than or equal to the thickness of the dynamic layer 6.

In the boundary layer both laminar and turbulent flow can take place. Accordingly, we will have a laminar or turbulent boundary layer, At flow rates gommon for a ZhRD, on walls of the chamber and nozzle a turbulent boundary layer will be formed. With this on

188

the main motion it is as if there is superimposed disordered pulsational motion, so that at every given point the flow parameters (speed, pressure, etc.), do not remain time-independent, and frequently and nonuniformly are changed along and across the flow. As a result there occurs a shift and mixing of particles of liquid (or gas).

These travelling particles are usually called turbulent moles or

moles of liquid, Although the magnitude of pulsations of flow parameters is insignificant and in all is several percent of their

mean values, the pulsations, have a decisive influence on processes

of friction and heat exchange.

Averaged and Pulsationsl Motion

Due to the complexity of the pulsational motion a strict theoretical calculation of parameters of turbulent flow is impossible, and therefore regularities of it are investigated for time averaged values characterizing this flow. Here the turbulent flow is examined as consisting of two forms of motion: averaged and pulsational,

Thus, if, for example, at the given point the time averaged value of the speed is equal to W, and the value of the pulsational speed is equal to w', then the speed at the given point during the time +

99+’, (4.8) where 5-е. (4.9)

Analogously, all the remaining flow parameters can also be represented in terms of averaged and pulsational values.?

Additional (apparent) turbulent frictional stresses and heat transfer

The presence of pulsational speeds affects the averaged motion in such a way that in it is as if the deformation drag is increased, i.e., there appears a certain additional, apparent, viscosity. Actually, when one examines in the boundary layer & line parallel

189

to the axis (Fig. 4,13), then the component along the y-axis of the momentum flow through a unit area per unit time

Леса. (4.10) In conducting the averaging of speeds, we will obtain J,=0(6+ w) (9+0')me(ae+o +50’ + we’). The mean value of the momentum flow will be equal to J,<0(e0+ew +o +a7)), (4.41)

Since according to rules of averaging

тыс somes, (4.12) then ет. (1.13)

That is, owing to pulsations the momentum flow is changed by the quantity eww. The transfer of the impulse component through the area parallel to the x-axis is equivalent to tangential stress; consequently, the magnitude of additional turbulent (or apparent) stress owing to pulsations

„= -we. (4.14)

Fig. 4.13. Momentum transfer with turbulent pulsations,

190

This streis has a positive direction along the x-axis, in which one can be certain of the following reasoning. Particles of liquid, which due to transverse pulsations enter intu the element from below (v' > 0), move from a region with averaged speed w and therefore cause a negative pulsation w', Conversely, particles, arriving into the element from above (v! < 0), 1.е., having larger W, Cause a positive pulsation w'; consequently, product w'v! always has a negative sign, and the additional turbulent stress Tryp фо

is always positive, i.e., it has the same sign as does the laminar stress

ap. 44 ие (4.15) The full tangential stress of friction + can be expressed in the

form of a sum of two components expressing laminar and turbulent friction:

=p >. (4.16) Together with the secondary stress of friction with turbulent

motion owing to pulsations, an additional heat transfer also appears,

Since mechanisms of heat transfer and momentum are similar, then, not repeating aforementioned reasonings, the expression for

the additional convection heat flow will immediately be written in the form

fonn™ — 6¥,9Te, ‚ (4.47)

where Tho — pulsation of stagnation temperature,

As is known, convection heat flow with laminar flow is equal to (when Pr = 1),

omen ene, Bm Se, (4.18)

191

The total heat flow, caused by laminar processes and turbulent

mixing, can be expressed in the form of the sum of two components (4.17) and (4,18);

пыле FO весть В gee, eT. (4.19)

Expression of secondary stress of friction and heat transfer through the length of the path of mixing

By analogy with molecular friction the secondary stress of friction

= ния = $. (4 e 20) where € = им — apparent kinematic viscosity.

To calculate e Prandtl proposed the following simplified scheme of flow, If one were to return to Fig, 4,13 and examine the two layers of liguid at distance Ay, then the turbulent mole, travelling due to pulsation from one layer into the other, maintains at a certain distance the component of momentum in the direction of the x-axis,

This distance, at which the mole of liquid preserves its properties in the direction of the x-axis, is called the length of

path of mixing and is designated usually 2. Thus, if 4y = 1, then particles proceeding from the lower layer into the upper preserve the horizontal component of speed W,. The difference between the average speed at point Yo and average speed of particles proceeding here from the lower layer will give in this place pulsation of the speed

Н 118 вша11, then Aw =lds/dy, and then > oo =/ —., 4.22 = (4.22)

192

Considering pulsation v' = w', we can express Ттурб in the form

tom 77 man SY, (4.23)

Similarly, the length of the path of mixing can be expressed as быти. Assuming in equation (1.17) сы 08 /Юу; Ти >0ТыЮу, ипеге 1 1 18 the length of the path of mixing for energy, and considering 1 ~ bas we obtain

Чл — бк Тот бк = Ta, (4.24)

Structural Diagram of the Turbulent Boundary Layer

A structural diagram of the turbulent boundary layer is shown in Fig. 4.14. The initial section of it 1s laminar. With an increase in thickness of the layer, the laminar layer is turned into a turbulent layer, The limiting thickness of the layer te» and length of the laminar section лам Черепа on parameters of the boundary layer, M number, state of the surface of the wall and critical value of the Re number

we Fig. 4.144. Structural diagram of the turoulent boundary layer: 1 — laminar sublayer; 2 — turbulent layer,

Under appearing turbulent boundary layer near the wall there remains a thin layer in which the laminar flow is preserved, This layer is called a laminar sublayer.

The thickness of the laminar sublayer Snay 18 determined by the critical value of the complex

193

ofan Santana | (4.25) Paw МПеге чаш, фыш €Nd pee — speed, density and viscosity, respectively, on the border of laminar sublayer.

For an incompressible liquid the experimental value a = 11,5,

In examining equations characterizing the magnitude of fractional stress (4.16) and convection neat flow (4.19) in reference to the given diagram >f the boundary layer, let us note that in the turbulent part of the layer the value of the first terms in the equations shown is small as compared to values of the second components,

And, conversely, in the laminar sublayer the magnitude of the frictional stress and convection heat flow is determined only by first components, since here because of the absence of turbulent mixing second components lose their meaning.

Calculation of the Effect of Dissociation

As is noted above, in the turbulent boundary layer heat is transmitted due to the shift in moles from sections with high temperature into colder layers. The process of heat exchange in conditions of the chamber of a ZnRD is complicated by the fact thet heat radiation toward the wall occurs from the high-temperature flow of dissociated combustion products. Therefore, with movement of moles not only physical heat is transferred, but also the reserve or energy contained in these dissociated combustion products. With the cooling of the mole in them reactions of recombinations occur, which are accompanied by an additional liberation of heat. This even more intensifies the process cf heat exchange. If it is considered that in the boundary layer the composition of combustion products remains balanced then the total amount of energy transferable by the mole can be estimated by the magnitude of the change in full heat content of this mole in extreme points of its path, Thus, in the presence of chemical reactions the role of the temperature will _be played by the total heat content оо.

194

The expression determining the total convection heat flow, Caused by laminar processes and turbulent mixing in the presence of chemical reactions, will be written in the form similar to expression (4,19):

i> i ge ten, (4.25 ) 0 &

where Цоо -— pulsation of full heat content of stagnated flow,

The presence of reactions of dissociation or recombinations, which follow after a change in temperatures and pressures in tne boundary layer, greatly hampers the problem of avereging of the equation of state and quantity la=la(T, 9), since the value of the gas constant R depends on the composition of products of combustion, which 1s itself a function of temperature and pressure. Therefore, with the averaging of the equation of state it is necessary to examine additionally the question of whether the composition of the gas, calculated with average temperature, coincides with the average composition of the gas at a given point of the boundary layer,

In order to simplify the problem V. М. Iyevley produce introducing the concept of effective temperature Тэф, which is determined from condition

Tome, (4.27)

where T and R — temperature and gas constant of the dissociated gas;

Вн — ваз constant of undisesociated combustion products of the fuel of the same initial composition,

Since, obviously, Ry < В, then Teg 2 a's

Thus, by using in the carrying out of all calculations a higher effective temperature Teg, we consider this to be the intensification of heat exchange in the boundary layer of the flow of dissociated products of combustion, which takes place owing to the flow of reactions of recombinations. Here the heat capacity

195

toy

fom ae: (4,28)

The effective stagnation temperature of the gas having temperature Teg:

т, (4.29)

where бр эф.ср — а\егаве heat capacity in the range of temperatures Теф-Тефоо e

4.4, Integral Relations of Energy and Impulses Tor the Boundary Layer. Initial Equations

Equations of the turbulent boundary layer with an axisymmetrical gas flow have the following form:®

1. The continuity equation и 9, (4.30)

2. The equation of motion

oy

ov вр, 10

+= -t+— =, (4.34)

where =p 2-3. (4.32)

3. The equation cf energy

gro “+ ee ent Ct, (4.33)

where

i» &

where То — total stagnation heat content of the core We consider Thoo to be constant

of W(x) and Тр.ст(х) are know, tion, and Тр.от(Х) is usually assigned,

\. The equation of state taking into account equation (4,27)

р Ta (4.34)

5. Equations expressing the dependence of viscosity and thermal conduction on temperature P=@O Ti) А, (Ти, (4.35) where С and C, = constants,

Here r — the shortest distance from the axis of the combustion chamber to a given point inside the boundary layer; wand v —

projections of averaged speed on the x- and у-ахе8; + — stresses from frictional force on an area prependicular to the axis; qs = convection heat flow normal to the wall; Thoo and Цоо — total

stagnation heat content and its pulsational value; x — distance

along the generatrix of the chamber and nozzle; y — distance along the normal to the surface,

The boundary conditions are the following:

1. When y = 0 (at the wall)

п о TymT nT Gees tty

2. When y =» 6

@=G; ta (¥ — speed in nucleus of the flow).

3. When у 2 A G0; Тел Ла

of the flow, along the chamber and nozzle; values

since W(x) is knom from the calcula-

197

= a 35

4. Conditions when x =» O will be examined below.

Integral Relations of Energy and Impulses for the Boundary Layer

On the basis of initial equations (4,30)-(4.35), let us derive the integral relation of energy for the boundary layer,?

Multiplying equations (4.30) by Ino, and (4.33) by г and adding, we obtain

Cetrolae) 4 Merle 200), (4.36) as oy Oy

Let us multiply equation (4,30) by the total stagnation heat content of the core of the flow T,,,.. Sine: Thoo does not depend on x and y, then

«ет гм. О, (4.37) Subtracting from expression (4.37) equality (4.36), and introducing designations 155 = Inog - Igqs To ™ Inoo 7 Ton (where Ion — total

heat content of gas at Thor and pressure p in the given section), we will obtain

М Vey ta) = — 2S. (4.38)

Let us integrate equation (4.38) within limits of the thickness of the thermal boundary layer A:

a é 8 Jar raw Oo tl + J a tree ata d= - i ay. (4.29)

With the boundary conditions accepted by us, the second term of the left-hand side of equation (4.39) is equal to zero, and the right side is equal to ВЧ т, whence

4 J + (reew e—Mwldy= RO,

198

or

Е (4.40)

Let us reduce equation (4.40) to a dimensionless form, for which

we will divide and multiply the left-hand side by corresponding characteristic values not dependent on y: the distance from the axis to the wall is R, the density on the boundary of the laminar Sublayer is eas, and the speed in the nucleus is м. Furthermore, both sides of equation (4.40) will be divided by Hsgo0 — coefficient of viscosity of gas at a temperature of the nucleus T3003 we obtain

ам ем. (4.41) The quantity fee 5-2) (4.42)

is customarily called the thickness of the loss of energy [79].

Let us give the Вер number in the form

Reyes (4.43)

Substituting expressions (4,42) and (4.43) into equation (4.41), we obtain

dz (RevRg) = Hie (444 )

Differentiating and multiplying both sides by УГО (where L — the arbitrarily selected characteristic dimension), we obtain

a i ; SPAR eats (5

where X = x/L; R= R/L,

199

Since we are free in the selection of a characteristic dimension, then we will subsequently take for it the critical throat diameter

D кр. Then in equation (4,45)

gut, Ra *.

2» ' Ow In order to reduce equation (4.45) to a more convenient form for integration, let us produce the dimensionless complexes:

- Dayco вах Regs мя. 1.16 2 ( )

=, (4.48)

where Poo and 00 — pressure and density of the stagnated core of the flow; Wess — highest possible speed of the core of the flow with expansion down to zero;

=== и <. та (1.49)

where бр эф-ср ~ average heat capacity in the interval of temperatures Ta -Тэфоо ®

Considering the characteristic dimension Dyp and substituting expressions (4.46) and (4.47) into equation (4.45), we will obtain the integral relation of energy

aRe i, , “Bit Re р Bae sf. (4459)

Conducting analogous transformations of equations (4.30) and (4.31), we will obtain the integral relation of impulses

Set Re Ley oh мени. (4.51)

200

Here Rey ~— Reynolds number obtained on the thickness of the loss of impulse $; Re, and $ are determined from expressions

Нм (4.52) Rep m= Sus , (4.53) ры complex ter Sao the dimensionless quantity emHy+1—(1—7,,) Hy, (4.55) where $ ltr у 4 lee A i, ту as (te) ae (4.57) Ser a a (4 58)

As we will Subsequently see, the solution to the system of

equations (4.50) and (4,51) under certain conditions can be reduced t. the solution of one of them,

For this we must set the Connection between quantities entering into these equations (Ат; Ан; Вер; Rey).

The Connection Between Вер, Ат» Ay» and Re 4

In order to find the connection between the indicated values, let us find the expressions for them in terms of quantities known to

us, For this it is necessary, in the first Place, to determine velocity and temperature (or heat content;) profiles in the boundary layer,

201 Ne a

In accordance with the structural diagram of turbulent boundary layer, let us examine separately the motion and heat exchange in turbulent and laminar parts of the layer.

Turbulent part of the layer. In examining this part of the layer, it is possible to disregard the first components in expressions (4.16) and (4.26). Then from these equations, by introducing the length of the path of nixing 2 according equations (4.23) and (4.24) and considering expressions (4,28) and (4.29), we will obtain:

teen epee С =. (4.60) where ha Tg Тел. (4 61)

To obtain velocity and temperature velocity it is thus necessary for us to assign this distribution 2, 1 and Qy across the boundary layer, so that it is possible to integrate equations (4.59) and (4.60). Calculations show that, although depending upon the assigned distribu- tion t, different dependences 1(y) and w(y) on the section of boundary layer are respectively obtained, and the numerical difference of the final results with different profiles of + is small, since the change in 2 as though compensates the change in tT.

Furthermore, profiles +t and qy in the turbulent part of the layer should be assigned in such a manner than on the boundary with the laminar sublayer they take the corresponding values tray and Чк.лам’ (Let us note that in the laminar sublayer it is possible

to assume tpay(¥) = CONST = tons Ix nay (¥) = const = Чет).

The law of the change in length of the path of mixing will be defined by the Karman function.

ime. у (4.62)

202

where » — empirical constant /» = 0.38-0.4),

Modifying the substitution of Dorodnitsyn, let us introduce the new variable

a= qi. (4.63)

Then the quantity of the path of mixing from equality (4.62) will be expressed as

bee —2g — 279 (4.64) Г. R Феи on where 1 c= ; (4.65) ou afer та жи) R Cun on?

Substituting expressions (4.63) and (4.64) into formula (4.59), we will obtain

tach ow. (4.65)

Considering considerations mentioned above on the selection of the profile +t and proceeding from conditions of the convenience of

integration of equations (4.66), the profile + will be assigned in such @ manner that the following equality will be satisfied

e Crew sou .

у. =, бе (4.68)

Owl °

(We place the minus Sign from considerations of physical meaning),

The integration of equation (4.68) gives

203

pagan

oe... 4, oy ath, (4.69)

where b ~ constant of integration,

Integrating expression (4.69) and determining the constant of integration from the condition that when w = W; n = 6, we obtain the equation of the velocity profile in the turbulent part of the layer:

Emits. . (4.70) where И, (4.71)

бт — thickness of the boundary layer in э1апе х-т.

Using expressions (4.63), (4.64), (4.68), and (4.69), from equation (4.60) we will obtain

a Хх fe. e Tones NS (4.72)

Let us assign now the profile dy in such a way that there will be satisfied the equality

fee e CyrQhQ® =p ng. sa Ora 8 aay

where c value of ¢ on the boundary of the laminar sublayer.

р эф.лам — р эф

Integrating equation (4.72), ме will obtain the equation of the profile of temperatures in the turbulent part of the boundary layer.

ати, (4.73)

where. 4, - thickness of the boundary layer in coordinates x-n;

пр (2eteele tan siaa Wo (4.74)

204

es

Let us turn to the consideration of the motion in the second (laminar) part of the boundary layer,

Laminar Sublayer, Let us determine the thickness of the laminar

sublayer Snaye For the case of an incompressible liquid it is deter- mined by the condition of Karman:

си _ ша, (4.75) where $. = У (4.75) = 11,5.

= ль лы Ве (4 TT )

where Велам — Reynolds number of the laminar Sublayer; es. — coefficient of viscosity on the boundary cf the Laminar Sublayer,

flow rate ® can be expressed by unit w

idth of the laminar sublayer in the form

Ma! �"". wen (4 . 7) From a comparison of expressions (4,77) and (4.78) Reng = =e (4 -79)

Expanding equality (4.79), obtained for an incompressible liquid, in case of the flow of 88s, we obtain

= @?, (4.80)

Let. us introduce а new independent variable 205

помни

EE м

Considering that in the laminar sublayer Tray 7 С"�" const and dx.nay 7 Gor = const, on the basis of equality (4.15) and (4.18) we can write:

их: (4,82) dex yt Se ey phe (4 .83)

Integrating equations (4,82) and (4,83), we obtain

= п. ме�' И ЧР 4.84 e lens 1; fens By aa! ( ) =——fo__ 4, ene ele : be роны п; ло�' в) р “=, (4 85) where 6 — thickness of the laminar sublayer in coordinates x-n;

тлам

"лам and toonam — on the boundary of the laminar sublayer,

Using expressions (4,81) and (4,84) and considering the initial equations (4,34) and (4.35), let us define

Tr Brow

4 М ыи Ци an = = Sey = Тем Ча To Vn | 7. }" m Гм и nen | 2 -п(- jan блич Gt nan.

(4.86) When n = 0.7 we assume (Te/T esau) ~ 1, and then from equation

(4.86) we obtain

былые в”. (4 87)

Substituting the obtained expression into equality (4.80) and considering condition (4.67), we wi’1 obtain

he (4.88)

206

син пои ить

In plane x-y the thickness of the laminar sublayer % Sy een f, a Mae ye I (re (4.89) Let us convert quantity Тэф/Т эф. лам entering into equality (4,80).

In accordance with expressions (4,29) and (4.61)

oe

Treat + fa A ——— —* a Te (4.90) че ва к

where cp od.cp ~ mean value of Cy эф in the interval of temperature

T3p7Too e

Dividing the numerator and denominator of the right side of equation (4.90) by Тефоо and introducing the designation

'-;- : ; (4.91)

А ® co-epf og

where & 9. cp — Value of ¢, og in the range of temperature Тф- and considering

Tap00°

соо ры рты, (4.92) after transformations we will obtain

fg __ font (Fra) B—w(S)

; (4.93) мона Че]

мпеге ТТ Рая ТР ж-Т,.. Let us define now quantities way /W;

оолам/ 00} оо’ оо entering into equation (4,93), From expression (4,84), replacing Top according to equation (4.67), and using equalities (4.71) and (",88), we will obtain

т... (4.94)

207

"nner en am

q

a a nas

— = -

Similarly, from equality (4.85), replacing Чст according to formula (4.74) and Snay 2cCording to formula (4,88), we obtain

wi a, From equations (4.74) and (4,85) we obtain ам (496)

Substituting now expressions (4,94), (4.95) and (4.96) into equality (4.93) and the obtained expression for 15$/ Тэфлам — into formula (4,89), using expression (4,88), after transformations we will obtain the final formual for 6,9, [quantity B°(w/#>> will be disregarded]:

8,,m Ob, (4 97)

where

ное ен] зави (я +1) (1 — Т.е) | 1+ (1—7, ri) ee — 2) |

Om (4.98)

Knowing now 65,,,,, we can determine the constant of integration b in equations (4.70) and (4.73). The length of the path of mixing on the boundary of the laminar sublayer

вин. (4 .99)

On the other hand, we find ‘л<�м from expressions (4.64), (4,68), and (4.69) when п=@, лам; [== вх; @= Qa, and considering r/R ~ 1:

Зита ны (5 sau +8). (4 ‚100 )

Solving equations (4.99) and (4,100) jointly with respect to b, taking into account equality (4.97) we obtain

bates (2-1) (1.101)

.

208

a

ae

or

Raf beds >. (4,102)

Quantity b < 6 лам’ and consequently, Ь < бапа д, Therefore, in sums (6 +b) and (A +b), which enter into expression (4.70) and (4,73), quantity b can be disregarded. Then, considering equalities (4,102) and (4.88), we obtain

$ auth 8 au? Фак eS ae ой)

Substituting value W, from expression (4.71) and considering that according to equality (4,35) Нлам == Мофоо (Тур. ла"�)", We obtain

Saw + Е. 80 (To ла"�)" (4.104 ) +5 ae RO, ' where Кери _ Оль, (4.105) Рю

By similar transformations we will also obtain

мы + $ aie $, ы® ‚ Фарлем ate (Ты =. =" Фа Аблам es г №, ° ( Bree )

where Rega Oe (4.107) Рф

Comparing formulas (4,105) and (4.107) with equations (4.53) and (4.43), which determine Вер апа Ве, we obtain

Ве = Ве, (4.108) Вер Ве. (4.109) 209

"7

Having velocity and temperature profiles in the turbulent part of

the layer and laminar sublayer, let us produce conjugation of the corresponding profiles by substitution of Wray/¥ from equation (4.94) and (5, sam +b)/(6 +b) from equation (4,104) into equation (4.70). After transformations

ФФ ные Ree 2 (4.410)

Similarly [see equations (4.95), (4.106), and (4.73)].

Rese м (4.144) 341.0 “М Values Top, nam and @yo, entering in equations (4.110) and (4,111) can be determined in the following way, According to equalities (4.29), (4.61), and (4.91), using equation (4.92), (4.94), and (4.95), the expression for T 5, nam can be reduced to the following form:

Ты = Т, „+ (1 = ет) un — (Pitan). (4 . 112)

Ехргезв1оп Гог алам сап be obtained from equation (4.65). Omitting the transformation, we will obtain the final dependence in the form

Can (4.413)

= Ъ`—ы + "а Pete |

With known or given quantities Thor and B, €, &, solving jointly equations (4.4112) and (4,113), Top. naw and a,,,, can be found by knowing them one can also determine Re, and Ве.

Using all dependences obtained above, we can turn now to the determination of quantities Ags Ан, Rep, Rey, and c interesting to

us in terms of quantities known to us and in terms of. @ and “т. According to the equation of state (4,34)

р fan _ бы 7, Comes ae Saat

210

Transforming now equation (4.47), taking into account expressions (4.114), (4.74), and (4.71), we obtain

Vie ant тым (4.115)

Similarly, from equation (4.54), taking into account expressions (4.414), (4.71), ana (4.57), we obtain

Ayp=

У

To determine Re, and Re, we will find preliminarily the

expressions for determining the thicknesses of the loss of energy 6 and impulse $,

Replacing in equation

(4.42) the variable according to expression (4.63), we obtain

веер an

where 4

т ~ thickness of the thermal boundary layer in coordinates X=.

Substituting into expression (4.117) values w/w according to

formula (4.70) and ‘оо/®оо according to formula (4.73), considering in view of the smallness of b that (6 +b) = 6 and (A+ b) = 4, and also assuming 6 = 4, we obtain

a 8 1 1 6. '- |" = |) (1.118) Integrating by parts, we define

+7 (1-2). (4.419)

Similarly, from equation (4,52) after transformations we obtain

+= (1-2). (4.120)

211

To determine Re, and Rep, we substitute into equality (4,108) expressions (4.110) and (4,120) and obtain

ух И), (4.121)

and into equality (4.109) — expressions (4,111) and (4.119), then

Fog matt 1-2), Regan = ; (4,122)

Similarly, determining preliminarily Hy and H, from equation (4.55), we will obtain the expression fur the dimensionless value

al! = +)-( ele . (4 aes

Thus, we obtained dependences (4,115), (4.116), (4.121), and (4.122), which express variables entering into the integral relations of energy (1.50) and impulses (4,51) in terms of quantities known to us and in terms of @ and т. Substitution of these expressions in equations (4.50) and (4.51) will lead to a system of two differential equations with two unknowns é@ and т.

The approximate solution of this system for the general case of motion in a boundary layer can be found; however, for practical calculations of heat exchange in a ZhRD it is more convenient to use the approximate but very simple dependence

The use of dependence (4.124) permits reducing the solution of the system of equations (4.50) and (4.51) to the solution of one equation,

4.5. Sclution of the Integral Relation of Thergy ace

Earlier we obtained the following integral relation of energy (4.50): .

212

3g

aR a(t (в.

Ft Re EP pe ee) = pL Examining expressions (4.115) and (4,122), which determine

into these formulas are functions Trews В, € and é.,, Consequently,

considering dependence (4.124) connecting ¢ and Gn» we can say that

Вер and A» are functions of only three quantities: Th oe? B, and Ems 1.е.,

с Ве = / 1 (Tren в, у ); Аг = } 2 (Teer в ty), (4 ° 125 )

The joint solution of equations (4.115) and (4,122) with the

assigned different values of Theor permits excluding €, and finding

the connection between Вер апа A, at different values of B, Expressing this dependence graphically it is possible to select the approximate formula of the connection between Вер апа Ат.

In the range of the change in Theor = 0-1; В = 0-0.8, апа lg Вер = 2.5-6

Аг = (141,5 Тата) п habe (4.126 ) 82 4,4 2.616 ( + 1415) (Fret + 0,08)

Let us introduce the following designations;

bpm 1+ 1,572. (4.127) ‘pain 4,14Rep = РИ li Tae) Fier t0.04.e | or

си (1 чем. в 0,04). ‚ (4.42 9)

Then formula (4,128) acquires the form

Rey=a,e*r, (4 .130)

213

i a вы

a на а-я

Let us substitute expressions (4,128) and (4.130) into the integral relation of energy (4.50) and reduce it to the form

ha (ere )+- Har ete RE) Пров. (4.434) Let us introduce into equality (4.131) instead of K and То quantities B/2 = FR and J Corgl vgn (t Тм) ед Рю Ть) ales Г, aly, (4 ° 132)

where c a7 effective heat capacity in the interval of temperatures (Trane Naboo)* We consider it to be constant along the length of the chamber, i.e., not dependent on x.

Substituting equality (4.132) into equation (4,131), differen- tiating and dividing everything by a,, we will obtain

adr +eperrataleD(!— Fin = SR PEE. (4.433)

For the convenience of the solution of equation (4.133), we denote

в"�". (4.434)

It is easy to be convinced that

баре“, (4.135) and. (fe? izzy, (4.136) where 1 cela ц. = :-# + Ей.

With the practical range of the change in y, within 7-15, the change in k,, corresponding to dependence (4,137) is small and umounts

214

to 1,32-1.15. Therefore, it is quite permissible to consider ky = const = 1,2, Substituting equality (4.135) and (4.136) into equation (4.133), we obtain

dtr + phy не О — Г, ах, r

or @ [epD Tel"? „№ в az м дат, ar ha

whence

4 lzp[e,D(1 — Т�")" = a’'~' [D(1 Рита. (4 . 128) r

Integrating equation (4.138), we find

‚. 1 М и _F ye - 4.1 fr omar | fe [2(1-7....)] ret de}, (4.139) where By — constant of integration

When xX = 0

Вр: [636 (1- Т.Е. (1.110) Determination of 2, in Conditions of a ZhRD

In reference to conditions taking place in a ZnRD, it is possible to mare the following simplifications of equation (4,139).

1. The constant of integration B, characterizes by itself the boundary layer on sections lying up to the beginning of the reading (i.e., up to the point where X = 0), The whole preceding analysis of heat exchange in the boundary layer in no way considered the influence of the mixing and burning on the development of the boundary layer. Therefore, strictly speaking, we should consider the beginning of the reading to be the section in which combustion wes completely completed, However, in this case we will have

215

remaining constant By, which we cannot determine, since we do not know the influence of processes of mixing and combustion on the development of the boundary layer.

In order to exclude the problem of determining By, let us assume conditionally that combustion occurs only in a certain narrow, "effective" flame front, lying immediately behind the zone of evaporation and mixing. With this, disregarding the initial section of the laminar boundary layer (may = 0), we consider that in the indicated section of the effective flame front there immediately starts the formation of a turbulent boundary layer, on the further development of which along the length of the chamber processes of mixing and combustion have no effect.

Taking now for the reading point the section where there starts to be formed a boundary layer, we therby exclude from consideratior. constant Br». since at this point Вт = О. In carrying out practical calculations, the position of the effect flame front, 1.е., the reading point X = 0, can be determined, considering that the length of zones I and II (see Fig. 3.2) consists of 25% of the length of the combustion chamber. The possible error in the determination of the reading point does not have considerable effect on tiie value of maximum convection flows in the nozzle and is therefore not dangerous,

2. Let us take the assumption about the constancy of temperature of the gas wali according to the length of the chamber and nozzle, 1.e., we will consider Trer™Tece/Tege™const. Such an assumption does not correspond to the real distribution of Тр.ст along the length of the chamber of the ZhRD; however, as we will see later (section 4.7), the error appearing with this in the determination of Чк can be easily corrected, and the assumption of constancy Tp.er permits

greatly simplifying the problem of the determination of Zn (and,

consequently, heat flows).

2. In accordance with equation (4,127) when Thor = const and b, = const,

216

4, From equation (4.129) we see that when Tot = const quantity ap Still depends on 8. However, since in expression (4.139) quantity a, is contained in power (К. - 1) ~ 0.2, then the effect of the change ineyr-'on the change in B is small and with integration of the right

Side of expression (4,139) it is possible to take also er?" w= const.

5. In the preceding analysis we assumed Pr = 1. For products of combustion of a ZhRD Pr = 0,75-0.8, Since this distinction of the Pr number from unity appear in the final determination of convection heat flow, we will examine the influence of tite Pr number on q» below,

Taking the assumptions noted above and considering that Веоо does not depend оп xX, formula (4.139) after transformations takes the form

ито ча. (4442)

Let us examine the obtained expression, As we noted above, X = x/Dyp» where x is the coordinate counted off along the generetrix of the contour of the chamber and nozzle. In the calculations it is more convenient to calculate with respect to the coordinate counted off along the axis of the engine xX, (Fig. 4.15),

From consideration of Fig. 4.15 it is obvious that

Gu. (4.142)

According to well-known formulas of gas dynamics (see, for example

[i]) entering into equality (4.141), quantities B, P/Poo and D are defined as

a&—! p=) atl

where =e — velocity coefficient; a f-=(i- my; (4,143) № 217

oh ly

k by the index of the isentrope Nyg and substituting equalities (4.442) and (4.143) into formula (4.444), we will obtain

t= ee frre De ti e

4. cosy

Calculations show that a change in Пиз does not have a considerable effect on quantity Zs Since the effect of the change in Пиз On the integrand expression and on quantity Вед, determined by expression (4.46), 48 the opposite. Therefore, with a sufficient degree of accuracy it is possible to assume Пиз = 1.2.

From expression (1.144) we see that with constant nig with one- dimensional motion in the nozzle quantity B depends only on the

ratio £/txp = (D/Dyp)s i.e., it is the function 5 = D/D yp. Accordingly, the complex

Тому) (4.145)

Taking into account relation (4.145) 1% 1в convenient to present the formula for the determination of Zp in the form

ote | Da, (4.146)

For the convenience of determining f(D) in Appendix т on Fig. I.1

there is given an auxiliary graph of the dependence of f(DB) on values DB,

As can be seen from expressions (4,127) and (4.129), the complex

sapr=/ (Fes D),

218

muthentinmeeeen ae ee

and therefore for easing the calculations Zq it 18 convenient to represent this dependence graphically (see appendix, Fig. 1.3, I.4).

Determination of z, for an Engine with a Cylindrical Combustion Chamber

In the determination of 2т for an engine with a cylindrical combustion chamber, it is expedient to divide the integral in the right side of (4.146) into two integrals, i.e.,

a 5 (+ Daz, чита, (4.147)

where X, — distance from the reading point ("effective flame front") to the end of the cylindrical section cf the combustion chamber (see Fig, 4,15),

Fig. 4.15. Determination of coordinate х..

If the chamber is not high-speed, then for its cylindrical

pert it is possible to assume Р/Роо = 1, B = const. Then the first integral of equation (4.147) is easily solved;

fr Daioh De, (4.448)

where By and Dy — values of 6 and 3 in the combustion chamber,

219

Say Se ЧИ re

Considering that zones of evaporation and mixing occupy 25%

of the length of the cylindrical pert of the combustion chamber, we assume

%,=0,75L,, (4.149). where

ne

Duy

Since in the combustion chamber B is small and 82 « 1, then,

disregarding in expression (4,444) value В2, when n

из = 1.2, ме will obtain

3 a Иня д. ео

Substituting equality (4.148) into expression (4.447) and using dependences (4,142), (4.445), (4.149), ana (4.150), we will obtain

in final form the formula for determining 2т for @ cylindrical non- high-speed combustion chamber:

aie aa] | (4.154)

4.6. Calculation of Convection Heat Exchange | In a АНХ ЕЕ

‘Using expressions obtained. above for R€ p05 2т, Vp» and Ans one can determine the magnitude of convection heat flows q . However, for carrying out the calculations the basic formulas in the dependence are expediently obtained in a more convenient form,

Determination of Effective

Stagnation Temperature Т5фо

0

Although the determination of effective

temperature in general is sufficiently Clear from its formal express

don (4.27), let -us give 220

the order of calculation of the effective stagnation temperature T p00 in reference to conditions in the chamber of a ZhRD, According to the indicated formula

Fogg = Si , (4.152) where �>> апа Ro — real temperatures and the gas constant of products of fuel combustion in the chamber, which are determined by conventional methods (see for example, [2], [25]); Rj, — ga8 constant of undissociated products of fuel combustion of the same composition,

In the determination of the composition of undissociated products of the combustion, the system of design equations is considerably Simplified, since equations of reactions of dissociation dropout, Thus, with the application of a four-element fuel with Н, 0, М, and C as a base, it is possible to cite the prepared formulas of determining the composition of products of combustion for a particular case when at a < 1 oxygen suffices for the full oxidation of carbon but is insufficient for the full oxidation of hydrogen. In this case the products of combustion will consist of CO., Но, Ho» and No. Calculation formulas for the determination of partial pressures of undissociated products of combustion will take the form

Poo, = a ; 1+6 Hetv, 3 Ne + wNy С, + С. 7 C,+C,

3 0,450, — Lie (4 С, + С. 2)

146 Hy tvHe 3 Np + iNe г Cr+vC, 7 Ci+vC,

(1.153)

рн.о =

H, + vH, 3 0,+ vO т eo «= OU, о rls С: + С. 4 C+,

Pu,= reg cette, 3 Nt WNe ' Се 7 Cp +vCy „3. № + У№ Рк, = : 7 C, + vC,

146 Het vHe | 3 № + № Ce+vC, 7 C,+vC,

221

Having determined the composition of undissociated products

of combustion, let us find for them Ry» and with respect to expression

Fig. 4.15. Dependences бр эф. ср» эф’ Th? А

and S on the relationship of components for fuel {80% HNO, + 20% N,0,) + Tonka 250,

222 :

Determination of the Value of Effective Heat Capacity бр эф

An examination of the afore mentioned formulas shows that contained in them are different values of бр эф: = 1n formula (4.49) in determining Wax it is the average in the interval of temperatures (T367T 000)» and in expression (4.90) aud dependences бр эф Tesulting from it is the heat capacity on the boundary of the laminar subdlayer. Since the difference in р эф is small, it is convenient in carrying out calculations to use one mean value of effective heat capacity, and therefore in further calculations we will use бр 20° taken at the mean temperature Top = (Ta@00 + Teo )/2:

Quantity Ср эф Can be determined Girectly by formule (4.28). To calculate the dependence бр ag On effective temperature, it is necessary to produce a thermodynamic calculation of the composition of dissociated products of combustion of the assigned fuel at different temperatures and calculate the appropriate values of effective temperature, after which, by using formula (4.28), 1% 1в possible to construct the unknown dependence (see Fig. 4.16).

Determination of the Coefficient of Dynamic Viscosity of Products of Combustion Tagoo

Entering into Calculation formulas is the coefficient of dynamic viscosity изфоо of products of combustion corresponding to temperature Т фо. It can be determined in the following way. By knowing the Composition of products of Combustion, we determine the coefficient

of dynamic viscosity p at an arbitrarily selected temperature T according to the well-known formula

sy

223

where Wy = viscosity of components of the mixture at the selected

p,m temperature (7), [25]; &, = Sa, — weight part of the component, where my — molecular weight of the component,

Usually value T is assigned within 1000-1400°K, since for higher temperatures there is no date on the viscosity of components of the mixture.

The found value y» at the assigned temperature T is calculated on the coefficient of dynamic viscosity at temperature Tep00 according to the initial formla (4.35) when n = 0,7:

Ream) (4.154)

Fig. 4.16 gives values of изфоо for fuel (80% HNO, + 20% №20, ) + + Топка 250,

Conversion of Formula for the Calculation of R€5o

According to equation (4.46)

Рен = еда бам ,

Pee

The density of the stagnated flow Poo according to the equation of state (4.34)

wn ae (4.155)

The maximum exit velocity by formula (4,49);

sn Fina в

224

where бр эф. ср - mean effective heat capacity.

According to the well-known thermodynamic relationship.

Substituting expressions (4.155), (4.156), and (4,157) into equation (4.46) and considering Nyg = 1.2, we will obtain

боль] забыт, „ох аль

Inasmuch as in the graphs and information tables the coefficient of viscosity is usually given in the form of и. 108 kets /m@ and pressure py, — in [atm (abs.)], the calculation formula (4,157)

can be thus converted: 1,107. к. Regen = -108) Re: oy (4.159)

or in the International Unit System

Ферри =3, - > , Ree Mogae у А.Т чи

= 2 -— 2 where Dyp in м, Poo in N/m“, Heboo in Nes/m“, R, in J/kg-deg and T5400 in °K,

Determination of Convection Heat Flow dx

By knowing Re,,, we can determine by formula (4.146) ог (1.151) the change in Zm @long the length of the chamber and nozzle of a ZhRD, and then by formula (4;134) —y,. Since in formla (4.134) у. is not expressed in evident form, then for determining ¥y it is convenient to use the auxiliary graph Ут = £(2_) (Fig. 4.17).

Substituting into equation (4,47) value Ay, expressed by Ут. and be from equation (4,128), and having replaced Тоо according to equality (4,132), we will obtain for the determination of convection heat flow:

225

Geese 6900.09 Fog (Fi. 0) Феб a e (4,160)

Let us replace Poo in equation (4.160) according to formule (4.155)

Р/Роо according to formula (4.143), and instead of speed W we will substitute its expression

И а (4.164)

tam!

and then we will obtain

tt. | «У / р-н (4.162)

Неге Чу 15 ехргеззей in keal/m@s and Poo ~ in кг”. Changing the dimension of dy and Poo and considering Nyg = 1.2, after calculation of the constant coefficient, we will obtain

и keal/m?h (4.163)

where Poo — in [atm (abs.)]

Fig. 4.17. Dependence ® №»).

$ Agee

Quantity A(1 - 82)6, as was mentioned above, is the function De D/Dyp- For the convenience of calculations in Appendix I on Fig. I.2 an auxiliary graph of the dependence 6(1 - A°)° » £(D) and auxiliary graphs of the dependence В апа (1 - 82) on D are given.

226

Calculation of the Effect of the Pr Number and Final Calculation Formula for Determining q

Formula (4,163) 1в obtained by us in the assumption Pr = 1, In reality for products of combustion of a ZhRD the Pr number is Pr = 0.75-0.8, and therefore to obtain the final calculation formula we ‘aust introduce into formula (1.163) the appropriate correction.

In accordance with formulas (4.2) and (4.47) and taking into account equality (4,132), the coefficient of heat radiation from gas to the wall a, is equal to

Gat tty

Considering 56 ~ A, the number Nusselt of the boundary layer will determine in the form

et. ad peg ae в Е New —- w= £"" ws 4.15 А Срок a Try Pr, ( 1 5)

where A and uw — coefficients of thermal conduction and viscosity of gas at a certain average temperature along the boundary layer. Substituting value a, (4.164) into equality (4.165) and replacing

Р/Роо DY берГербиТыю, Ме will obtain

Nome? py, (4.166)

where г. =. 7 Tem Introducing Ren, we get

Noe As is known, with motion in a pipe at low speeds (i.e., B) heat exchange is determined by a dependence of the form

(4.467)

227

Я

Мари f (Re, Ти).

In general, taking into account the effect of В оп heat exchange, it 1s possible to write

Num Pre (Re, Tren f: (4.168)

Examining approximately the motion in the boundary layer as the motion in a pipe whose diameter of which is proportional to 6, and substituting the value Nu from the dependence (4.168) into expression (4.167), we obtain

ere 4,169 + Ге. Тн И (#10) whence, considering that Top is proportional to Тр.ст’ and Re is proportional to Reg, we will obtain

Aras Pr™ f, (Ren Teer = (AF er=1 Pr. (4.470)

Comparing expressions (1.170), (1.160), and (1.128), ме obtain that, taking into account the influence of the Pr number

ee. (4.471)

Thus, a decrease in the Pr number of products of combustion leads to a certain increase in heat flow. In conditions of the operation of a ZhRD (Pr = 0.75-0.8) the correction for the influence of the Pr number is equal to 1.15, 1.е.,

9 == 16 (4.172)

Introducing this correction into the numerical coefficient of formula (4.163), we will obtain the final calculation formula for determining the convection heat flow:

tamer ip OPP sewer hoa (nen

ae (4.173)

228

or in the International Unit System:

¢,~3,0"" (19 Poet » 00.097 v000 (17 rer) W Иде, om

where

Poo in N/m, бр эф. ср and R,, in J/kg-deg, T p00 in °K,

If in the calculation of heat exchange in a ZhRD it is necessary to determine the coefficient of convection heat radiation from gases to the wall Qn, then with the known Gy it, in accordance with equation (4.2), will be determined by the expression

Effect of the Boundary Layer on Heat Exchange and Calculation Ч» with Internal Cooling

The aforementioned method of the calculation of convection heat transfer assumes that between the core of the flow and the wall there are no intermediate layers of gas with perameters different from parameters of the core of the flow. Such a case is fully possible when walls of the chamber of a СААР are protected by thermoresistant coverings applied to the internal surface of the walls so that the necessity in internal cooling drops. However, with the protection of the walls by internal cooling, between the core of the flow and the wall a protective boundary layer of gases of lower temperature will be formed,

As was shown, this boundary layer can be formed by both installing on the head special peripheral injectors and with help of feeding internal cooling through cooling belts. An analysis of the effect of the distribution of injectors on Vog and the method of calculation of the distribution of Yop along the perimeter of the chamber are given above in section 3.6.

229

ot aga

With the supply of internal cooling through the belts, very approximately one can assume that below them along the flow there will be formed a boundary layer with the relationship of components

ч=.бым+ 0

* Олевл + Отт ' where бр, охл апа охл — Р1ом rate of fuel and oxidizer for internal cooling; Gn ст and Go on flow rate and oxidizer through

the annular the wall whose thickness is equal to the spacing between the injectors.

With a boundary layer near the wall there will be formed a boundary layer whose parameter are determined already by parameters of the given boundary layer.

If the thickness of the boundary layer np is quite great as compared to the thickness of the boundary layer A, then one can assume that convection heat transfer in the wall occurs no longer from the core of the flow but from the boundary layer.

In this case the aforementioned method of calculation of convection heat flow remains acceptable under the condition that by effective stagnation temperature of the core of the flow Т5фоо is implied effective stagnation temperature of products of combustion Тэфоо» which is obtained with the relationship of components in the boundary layer Уст In conformity with these values Тэфоо апа Уст, it is also necessary to take quantities Ин» бр ab.cp 2nd Hoos

Calculations show (Fig. 4.18) that in the boundary layer temperature Teg is very close to the temperature of the undissociated gas Ty? determined also under the condition of constancy of full heat content; in the range of temperatures usually taking place in the boundary layer (1700-2200°K), in practice temperatures Tap? Ty and T coincide.

230

2989 ae me 6 ht

tee Fig. 4.18. Dependence of of T, Ty and Teg on heat

content for a fuel of oxygen +95% ethyl alcohol.

Proceeding from this, in Calculations by definition of convection heat flows, with internal cooling the effective Stagnation temperature of the boundary layer can be taken as equal to the temperature of products of combustion with given Уст of the boundary layer,

Depending upon the distribution of Yoo along the perimeter of the chamber, distribution of temperature of the boundary layer and convection heat flow along the Perimeter -f the chamber will be changed accordingly,

Figure 4.19 gives grarhs showing the effect of the change in Уст Along the perimeter of the cross section of the chamber on the magnitude of convection heat flow Ay

The calculation of the cooling of a ZhRD, obviously, should be conducted under the most difficult conditions for the wall. This means that in the case of a screen of cooling with a surplus of fuel (aon < ядра) the computed value Voq Should be the largest of its values along the perimeter of the chamber of the engine, With this, obviously, in places with smaller values of Vo Convection heat flows and total heat removal from the surface of the chamber and nozzle of ZnRD will be less than their computed values determined

231

with the greatest (1.e., the worst) value of Уст. But this inaccuracy will proceed as if it is in the safety margin of cooling of the chamber,

Fig. 4.19. Effect of the change in Уст along

the chamber on quantity Чу.

Order of Calculation of Convection Heat Flows

Let us assume that geometric contours of the combustion chamber and nozzle are known. The fuel and relationship of components in the core of the flow or in the boundary layer are also known (in case if there is a boundary layer). Then the calculation of convection heat flow can be conducted in the following order;

t. Let us divide the chamber and nozzle lengthwise into separate sections, The nozzle part, depending upon the expansion ratio, is usually divided into 12-20 sections. The combustion chamber is divided into 3-4 sections lengthwise, If, however, the combustion chamber is of cylindrical form, then sometimes tne entire part of the chamber from the effective flame front up to the beginning of the narrowing part is considered as one section. For every section we determine the average dimensionless diameter BD and average angle of inclination of the generatrix y, which for the given section are considered constant,

232

e. Let us assign the distribution of temperature of the gas wall Тг.ст along the length of the chamber and nozzle of a ZnRD. For the simplification of calculations, usually in the first

approximation we consider Тр. ст constant along the length of the chamber and nozzle,

Here it is convenient to take the value Ther as the average between the maximum allowed material of the wall by value Theor an expected value Treo in the chamber or on the section, For steel walls the mean value Тр.ст = 600-800°K; for walls of copper or alloys with copper as a base the average Troon = 400-700°K,

a

By determining the values of heat flows with Theor constant along the length, we determine the final values of dy at the assigned distribution Тг. ст along the length of the wall with the help of the conversion formula (see section 4.7),

If in calculating x it appears necessary immediately to assign the distribution Theor along the length of the chamber, then values and the character of distribution Theor in the first approximation are selected according to recommendations discussed later (see section 4,42),

3. Let us determine Taboo? Hegoo? бр 96.cp? and R,. The order

of determination of the values shown is stated above, In the

presence of a boundary layer values Тофоо, афоо? бр эф. ср and Ry are taken not for the core of the flow but for parameters of products of combustion with the relationship in the boundary layer Vor Determining Taboo: we find the distribution along the length

Ther = г.ст/Тэфоо and necessary for further calculation auxiliary quantities by, b=, and цен

4, According to formula (4.159)

2 Da en so Geer OV Sree,

Fer the isobaric chamber Poo = Ро; for the high-speed combustion chamber Boo = Р200.

233

5. Let us calculate the quantity Zp along the length of the chamber and nozzle, According to formula (4.146)

apm ie a,

In the calculation of Z_ With a cylindrical combustion chamber let us use formula (4,151);

vale [fi]

In the given formulas quantities t/arbt D'* and f(D) are found by auxiliary graphs (see Appendix I, Fig, Т.1-1.4).

6. On the auxiliary graph (Fig. 4,17) let us find the change in ¢r=(f(#r) along the length of the chamber and nozzle.

7. By knowing y,, according to formula (4.173) let us determine for every section

099 Pent » op.ert come (—Frer) 0 450: 10° . . ag Vidun =

We find value A(1 - 82) by the auxiliary graph (see the appendix, Fig. 1.2).

Determination of Convection Heat Flows by Criterial Equations

As we see, the calculation of convection heat flows on the basis of the solution of a system of equations of a turbulent boundary layer is a very laborious problem, Therefore, sometimes for obtaining the distribution of specific convection heat flows along the length of the chamber of the engine criterial formulas are used, ‘

234

To determine Gy the application of the criterial formula of Gukhman-Ilyukhin [73] 18 the most widespread:

Num=0,0162Pet (Ta)

After the substitution into this formula of values of criteria, we will obtain the following formula for determining the coefficient of heat transfer from gas to the wall [25]:

в == 74,Зс р (врем я keal/ mh “deg (4.474 )

Tea or in the International Unit System: 6, —0,0006¢, nt" Soe (ta) W/m-deg, ner

where ¢, er kcal/kg-deg (or J/kg*deg) and Buon kef/mes (ог Ист №. 8/2) — heat capacity and viscosity of products of combustion at the temperature of the gas wall; G — fuel consumption in kg/s; D —

diameter of section in м; Тоо and Troe — stagnation temperature

of the flow of products of combustion and temperature of the gas

wall in °K,

By knowing Qn, we determine the specific heat flows by formula

Ч. =а, (Ти— Тьет).

The order of calculation Gy by criterial equations and ап example of the calculation are discussed in detail in work [25],

4.7. Conversion of Convection Thermal Flows

As can be seen from the preceding, the calculation of convection heet flows is a laborious problem, Therefore, considerably more convenient is the method of the conversion of data according to convection heat flows for the "base" engine well-known to us into a projected one,

To obtain formulas of the conversion, let us find the approximate connection between Чк for two geometrically similar chambers of a ZhRD, which are different in dimensions, pressure in the chamber,

235

form of fuel, and relationship of components and temperature of the wall, Let us examine the expression (4.146). For geometrically similar engines in similar sections (i.e,, in sections where Б

are identical) the complex

о д (4.175)

will have ап identical value.

In formula (4.175) quantity 2, 18 connected with y, by the dependence (4.134). Approximately

(f= 18,82".

Substituting here value 2 from formula (4.175), we will obtain

dha 160 oe (4.176)

Substituting expression (4,176) into equality (4.173) and replacing R€og by formula (4,159), we will obtain the following dependence for the determination of qx?

а-муди аорте Уи, п я 7". (4.477)

<7" 1 315,05" . chao

The quantity os)" entering in formula (4,177) can be approximately represented in the form

oP) at”

Substituting this expression into formula (4.176) and uniting the constant factors into constant К. , we will obtain

=k, BP a6 wp.an'(T epee —To.ex) (ingen: 100%"

Pa ene — (4.178)

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Comparing the convection heat flows of the "base" (index I) engine and the engine geometrically similar to it, we will obtain for Similar points, in which DB and, consequently, complex B are identical, the following relation:

: Oar ОГ 69 „р. с Ре — Тат) р"�" ИР из ¢%8

= Dep" (Re pe ee (Re) ye) 4.179

al) (ee) Ue) ES mae) (ee) Gey бл Introducing in it the designation

Su: 8p cg-en Toque — Tra ) vo , ’ (4,180)

v tar

Quantity S depends on the kind of fuel, the relationship of components and the temperature of the gas wall, For each fuel it is possible to calculate auxiliary graphs of the change in S depending upon Theor and the relationship of components vy, For certain fuels such graphs are shown in Fig, 4.16 and in Appendix II (Pig. 11.1-11.5)

Formula (4,181) is obtained fcr geometrically similar engines, However, it can be used also for approximate calculations of engines geometrically not similar but similar in form, With these values of dy and S for both engines must be taken in sections where Dare identical, It is convenient to use formula (4,181) also with the necessity of the conversion of heat flows in the same engine but at different values of Тр.ст In this case, since Poo and Dep remain constant, the formula of conversion has the form

ФМ! =5/5',

I

where Ч»

апа st — values at some rated temperature,

237

4.8. Determination of Radiant Heat Flows ee eng OD

Radiant Heat Exchange in a ZhRD

In the chamber of the engine radiation of Products of combustion occurs. As is known, the radiative and absorbing ability of gases entering into the Composition of products of combustion is different; the greatest radiative and absorbing ability is possessed by

polyatomic gases, in the first place, water vapor Н20 and carbon dloxide CO,.

The radiation and absorption of gases has the following basic distinctions from the radiation and absorption of solids,

Solids radiate and absorb energy of all wavelengths from А = 0 to} = @ and gases — only in definite intervals of wavelengths (so-called bands) which are different for different gases. Such radiation or absorption is called selective,

Solids in their Majority are Opaque for thermal rays, and one may assume that their radiation and absorption occur in the surface layer, In gases radiation and absorption occurs in a volume. Here the radiative and absorbing ability of the gas are determined by the quantity of molecules of gas occurring in the volume, At a given temperature the quantity of molecules of gas is proportional to the partial pressure of gas p and volume of gas characterized by the length of the path of the ray 2. The average length of the path of the ray 1 depends on the form of the volume occupied by

the radiating gas. Table 4,2 gives values of . for gas bodies of different forn,

According to experimental data for gases, the radfation and absorption of water vapor are proportional to т? and of carbon dioxide — proportional to 72.5. Thus, for case of radiation of the volume of products of combustion in industrial furnaces the following - empirical formulas for determining the radiant heat flow are obtained [5]:

238

taom dip lio (i5)*— (ha) (4.182) бро, ЗУ рой (= : —")").

where partial pressures p and pry. are taken in kef еп? and the H50 605

length of the path of the ray i inm, Here the total radiant heat flow

Чун Фо, . (4.183)

These formulas can be used for a tentative estimate of quantity

Чл in the combustion chamber and nozzle of the engine, However,

at high pressures taking place in the combustion chamber of a ZhRD, the use of emperical formulas (4.182) insufficiently admissible, since they are obtained at atmospheric pressure and temperatures up to 2500°K, Therefore, a more strict method of the calculation of Gy, in a combustion chamber will be given below,

Table 4.2, Length of the path of the ray for gas bodies of different form, в в

Form of the gas body 1105

“Sphere with diameter Dy; . 8peric combustion chamber 0.6

with diameter a: | ylinder w 1атефег Dy,

infinitely long; radiation os on the lateral surface, yiinder w ‘Length В = Ох; radiation on the es lateral surface,

Cylindrical combustion

oe . 3—3 0.0

>4 | 0,9

For convenienc the basis of calculation of the radiation of gases is the Stefan-Boltzmann law. Here the radiation and absorption of gas, referred to a unit surface of volume occupied by, are taken proportional to the fourth power of absolute temperature,

239

In general the radiant heat flow in the chamber of the engine from gases to the wall is determined by expression

4. = чи 57) вт Ао (м) , (4.184)

where Tp and Тл.ст — temperature of products of combustion and gas wall, respectively, in °K; Eon. eh -— effective degree of blackness of the wall; €, — degree of blackness of products of combustion; Co = 4.9 keal/m°h- deg’ (5.67 W/m?- deg") — radiation factor of an ideal blackbody; Ap — absorbing ability of gas at a temperature equal to Th an.

The first term of equation (4,184) determines the radiation from the gases to the wall and the second ~ radiation from the wall to the gases. The order of the determination of Ест. эф, Ens and Ap will be examined below,

Formula (4,184) should be used in the calculation of qd, in engines having high values of Тр.с<�› for example, in the application of various kinds of ceramic or other heat-resistant coverings, In engines with copper or steel cooled walls not having any special heat-resistant coverings, Troot 1s comparatively small, and therefore the role of the second term of (4,184) 1s also small; and, consequently, the radiation of the wall can be disregarded,

In this case formula (4,184) will have the form

чин (1). (4.185)

Since the magnitude of radiant heat flow is determined in the first place by thermodynamic temperature, along the length of the nozzle a sharp lowering of values of Q alweys takes place, Figure 4,20 shows a typical Giagram of the distribution of Чл along the length of the chamber of the engine. Into the entrance part of che nozzle there occurs а sharp drop in values of Gy» and in the super- critical part they are negligible as compared to values of Чл. х in the combustion chamber, Therefore, in calculations of radiant heat flow it makes no sense to determine Gq for every section of the combustion chamber and nozzle, and by finding values of Qz.x in the

240

combustion chamber, it 4s possible with a sufficient degree of accuracy to take the following distribution of Gy, @long the length of the chamber and nozzle; near the head dy ~ 80% of the value of Чл.к› on the initial section of the combustion chamber with a

length of 50-100 mm ‘qy increases up to the value of Чл. х and further remains constant up to the section of the subcritical part of the nozzle, where D = 1.2Dyp. In the critical section q, = 0.54543

in the supercritical part of the nozzle in the section where

Da» 1.5D xp, Чл = Oday wand in the section where D = 2.5D xp

Чл = 0-24, к.

qs Fig. 4.20. Ddstribution of Чл along the length

of the engine.

Determination of ест. эф’ Ep and Аг

Values of Con. od? En, and Ay depend on the degree of blackness of the gases, components of products of combustion, and also on the degree of blackness of the wall of the chamber, Let us examine the method of the determination of these values given below,

Determining the degree of blackness of products of combustion Ег. As was shown, from the number of gases comprising the products of combustion of a ZhRD, only the radiation of H,0 and CO, is of practical importance for the calculation of Чл. Therefore, it is assumed that the degree of blackness of products of combustion Е. depends on the degree of blackness of water vapors Ep ¢ and carbon dioxide со, and is determined by expression г

ино ко, — тоже. (4.166 )

The last term in expression (4.186) means that the radiation of the mixture of H,0 and CO, 18 somewhat less than the sum of the radiations

241

of these gases, since the bands of radiation end absorption for Но0 and co, partially coincide, Therefore, the energy of the radiation,

for example, of Но0 is partially absorbed by the carbcn dioxide, and conversely,

The degree of blackness of water vapors н.о is a function of

temperature, parameter (p 2) and pressure in the chamber, Taking into account all these factors ЕН о 1s determined by formula 2

дм ал earn, (4.487)

where wq,o= SIT. (oyu gM — degree of blackness of H,0 when Ро О and

Po = 1 [atm(abs.)]; bnp= flr, ou coefficient taking into account the effect of pressure on Ен_о 2 ®

The value of sn,o=fT, (sn,0f} is determined from graphs of Fig. 4,21, obtained by means of extrapolaticn of data of Hottel and Egbert

(84). The dependency I+hne=fipn clon!) is represented on Fig. 4,22

(the effect of temperature on КН o 18 insignificant and is therefore not considered), 7

To calculate 1.9 by formula (4,187), 1t 1s possible to use 2 an auxiliary graph of the dependence но] (ци ойнюрн,о) (Fig. 4.23).

The degree of blackness of carbon dioxide to, f(T. (peo) is determined from graphs represented on Fig. 4, 2% (=со depends 2

insignificantly on pressure),

Determining 1,0 and C0,’ by formula (4,18) let us find the degree of blackness of products of combustions с.

242

aor у

Fig. 4.21,

D ear ependence fon,0 “AT, (Pu

4

1 42 4 Puse ker /on®

243

tue

0 4 fen,e

Fig. 4.23. Auxiliary graph Tor the calculation of

«н,о=Лон,о, #н,о; Ри,о) *

Fig. 4.24. Dependence ecosAT. (Pco,!)! «со, = ЛИТ. (Рсо,П1 >

Determination of effective degree of blackness of the wall Ест эф In the presence of a radiating gas the effective degree

of blackness of the wall Ест. эф is greater than the usual value of the degree of blackness of the wall e,, and stiil depends on the

degree of blackness of the gas Ер: fer vg ™ Fer (1 + (1 —2.,)(1—2,)], (4 .188)

where e, — degree of blackness of products of combustion (4.186); Ecp — degree of blackness of the wall with respect to the radiation

244

of gases and depends on the material of the wall and state of its surface. Since frequently the walls are covered by carbon black, then in this case independently of the material of the wall, it is possible to essume ест = 0,8, If, however, the carbon black on the walls is absent, tentative values of €or for different materials can be taken from Table 4.3,

Table 4.3, Degree of blackness Ест ГОГ

different metals,

Materials fee Aluminum oxidised at 600°C 6,112.19 Polished bronse @! Rough bronse 0.8 Tungsten а

Brass oxidised at 600°C

9,61—9,0 Copper oxidised at 600°C Mo.

ofeREEEeHtet |

lybdenum 0,90—0,13 Oxidised nickel 0.479, Miobium ez Alloyed steel $6.26 Oxidised steel 68 Steel greatly oxidised 0 Ground steel 99-1110 9.5-4,61 Titaniun #155 6,3-0,4 Chromium nickel 19—94 0,64~0,8 Cast iron oxidised at 600°C 208—905 6,649.96

Determination of the absorbing ability of gas A,. Since the temperature of the gas is not equal to the temperature of the wall, and since the gas absorbs selectively, then in general eA, However, with technical calculations it 1s possible to take A-=e,’, where en - degree of blackness of the gas taken at а temperature of the gas

wall Ty og. With this ef is determined by formula (4.186),

Determination of Gd, in the Absence of a Boundary Layer

In the calculation let us consider known the dimensions and form of the chamber, the composition and temperature of products of combustion, the pressure in the chamber and the temperature of the

245

бас ма11. Having these data, it 18 possible to establish the following order of Calculation Чл.

1. According to Table 4.2, let us determine the length of the path of the ray 2,

2. Let us determine by formla (4.186) the degree of blacknese of gas in the combustion chamber Ene

3. Considering Ест = 0.8 in the presence of carbon black on the walls or determining Ест &Cccording to Table 4.3 in the absence of carbon black, let us find the effective degree of blackness of the wall for.ep from formula (4,188)

4, In the case of a high value of Тр.ст 1е% us determine A ;

5. Let us determine radiant heat flows in the combustion chamber Gg.x° With ceramic or heat-resistant coverings, which cause high Tp.on» let us calculate 3.x from formula (4.184). In the absence of Coverings the radiation of the wall will be disregarded, and we calculate Qz,x% from the formula (4.185).

6. Knowing Qg.nxe let us determine the distribution of radiant heat flows along the length of the combustion chamber and nozzle of a ZhRD,

Determination of Gq in the Presence of a Boundary Layer

The boundary layer of products of combustion with internal cooling of walls of the chamber of a ZhRD has a lower temperature than does the core of the flow. This layer plays the role of a semitransparent screen between the core of the flow and the wall,

The energy amount radiated by the boundary layer itself is comparatively low due to its low temperature, At the same time this layer can absorb a Considerable part of the radiant energy proceeding from the core of the flow to the wall since in the

246

composition of components of the layer, just as in the composition of products of combustion of the core, the basic radiating and absorbing gases Но and CO, predominate,

Thus, the boundary layer lowers the radiant heat flows in the wall of the chamber of the engine by 30-604,

Let us examine the simplified method of the calculation of the influence of the boundary layer.

In usual limits of the relationship Yor/v = 0.3-0.6, the decrease in intensity of radiation into the wall can be taken as

constant, and then the expression for the approximate calculation of qy will be

ТИР (4.189)

where qy — speciric radiant heat flow in the absence of internal cooling (4,185); Cor.e@ — effective degree of blackness of the wall; 0.65 — coefficient con3idering the decrease in intensity of radiation with internal cooling; @ — coefficient considering the decrease in intensity of radiation of “he core of the flow because of a decrease in the surface ci the radiating volume of products of combustion

as compared to the surface of the combustion chamber absorbing the radiation:

gan aah

or, (4.190)

where Dy — diameter of the combustion chamber; H — spacing between

the injectors; Lim length of the combustion chamber to the critical section,

Thus, having determined the magnitude of the radiant heat flow in the absence of internal cooling in the order mentioned above, we find the final value of the radiant heat flow taking into account the boundary layer Gg.ct by formula (4.189),

247

4.9, Determination of Heat Transfer from the Wa о e Cooling Liqu

Heat is transmitted from the wall to the liquid coolant by means of convection. The magnitude of heat flow from the wall to the coolant is determined by expression

46, (Tan—Ta) (4,191)

where ay — coefficient of heat transfer from the wall to the liquid; Tg.0r 7 temperature of the liquid wall; Ty — temperature of the liquid,

The coefficient of heat transfer from the wall to the liquid Gy is most expediently determined by using the criterial equations obtained as a result of the analysis of experimental data on heat exchange in channels cf a different section. In reference to heat transfer in the coolant passage of a ZhRD with high heat flows the application of the formula of M. A, Mikheyev [84] 1s expedient and quite reliable:

Nu ==0,021Re®* Prt, (4.192)

In the given formula the determining temperature is the temperature of the liquid coolant Ty, and the characteristic dimension — equivalent (hydraulic ) diameter of the coolant passage dy.

Substituting into formula (4.192) the expression of Nu, Re, and Pr, we get

Stam qm (USS) (=) (4.193)

where a, meee of heat transfer from the wall to the liquid in keal/n* *8-deg (м/а2 deg); Mg — in keal/m-s-deg (W/m-deg); gig — in kg/m-s (or ug in Nes /m= )s Gy — in kcal/kg-deg (J/kg-deg) and уд — in kg/m, respectively, are а conduction, viscosity, heat capacity and density of the liquid coolant taken at the average temperature of it on the given section; dy — equivalent diameter of

248

the cross section of the channel in m calculated from expression

40%, (4.194 )

where fy — area of cross section of the channel in m°; ll — total (moistened) perimeter of the section independently of what part of this perimeter participates in the heat exchange; 6 — coefficient considering the direction of heat flow and the temperature difference (АТ Тая Та) .

Experience shows that with the heating of the liquid, i.e., in the direction of the heat flow from the wall to the liquid, the intensity of the heat exchange is higher than with the reverse direction of the heat flow, i.e., with the cooling of the liquid, Furthermore, the intensity of the heat exchange also depends on the temperature difference, 1.e., on the value Ty.or -Ty. With an increase in the temperature difference ‘he coefficient of heat transfer ay with the heating of the liquid increases and with cooling decreases,

The dependence of heat transfer on the direction of the heat fiow and the temperature difference is conditioned by the fact that the field of temperatures and viscosity in the boundary layer and thicknesses of the actual boundary layer with the heating and cooling of the liquid are different, This dependence is considered by the coefficient

ета (098

where far Cuer Amer — Viscosity, heat capacity and thermal conduction of the liquid coolant, respectively, at the temperature of the liquid wall,

For conditions of cooling of a ZhRD quant .y 6 is changed within limits of 1-2,5,

249

Let us reduce the expression (4,193) to a more convenient form for calculations

чтоб “ ow ee eof? (5.196)

According to the equation of continuity

where Goxn — coolant flow rate per second through the cooling

mane in ke/s; ty — area of cross section of the coolant passage in в.

Substituting expression (4,197) into equality (4.196) and

designating we aint (4.198)

we obtain

мы, Кса1/п?.в. Чек.

и upon the eerree system of units a, will Be obtained in koal/n* s‘deg or w/in? edeg. To determine a, in Кса1 /п2 hedeg frequently this formula is used in the form

a 862 43m)" keal/m@h-deg. (4.199)

Complex Z, as we see from formula (4,198), characterizes the physical properties of the coolant and is a function of the temperature. Values of complex Z for different components are shown in graphs of Fig. 4.25, where also values of the product of complexes ZB are given for different components obtained when Ty oe = 200°C.

250

0,26

0,18 01 0, 0,06

6 7

0 WO 60 200 Tec 0 100 150 209 Ty °C

Fig. 4+.25. Value of complex Z: 1 — H,0; 2 — 80% HNO, + 20% Np0,; 3 — 75%

CoH. OH; 4 — 96% CoH. OH; 5 — 6% HNO 3 6 — Tonka 250; 7 — kerosene; 8 — HDMG.

4,10. Forms of Coolant Passages of Chambers of a ZhRD

Figures 4,25 and 4.29 give difierent forms of coolant passages of the combustion chamber and nozzle of a ZhRD.

Slot Channel

The simplest coolant passage is in the form of a smooth slot channel (Fig. 4.%a); however, its main deficiency is in the small rigidity of the inner shell of the chamber. his can easily lead to the distortion of dimensions of the coolant passage and, as a result, to a change in the coefficient of heat transfer Qy and

burnout of chamber.

Furthermore, with small quantities of coolant and the necessary speed along the coolant passage, the width of the slot De is obtained very small (0.8-1.5 mm) and technologically almost not

feasible.

To increase the rigidity of the internal wall with slot channels and also for facilitating the manufacture of the coolant passage, between the wall and the jacket of the chamber of the engine inserts of plates or from wires are placed (Fig. 4.%b). The inserts as if calibrate the dimensions of the channel and, as a rule, se e also for the fastening of the internal and outer shells. With chis a

258.

so-called braced shell of the chamber of the engine possessing high strength properties is obtained, However, the installation of inserts is a very laborious operation, and therefore they are usually

used for calibration of the duct or the fastening of shells only in separate places of the chamber of the engine.

772 РДС > ee

2221. WEE er

„>=> SESS SoS SS

fe» § ЕР:

A

Wena rrr ГАА АСЯ

г) Fig. 4.26. Forms of coolant passages: a) smooth slot channel, Ъ) slot channel with inserts, с) slot channel with forgings, d) coolant passage with ribs, e) coolant passage with screw channels, f) coolant passage with corrugations,

A variety of the slot coolant passage is a slot channel with forgings (Fig. 4.26c). Here in the external shell of the chamber of the engine there are forged (usually in checkerboard order) round or oval deepenings — forgings on which by spot welding the internal and outer shell are fastened, With such a method of fastening there is obtained a sufficiently durable and rigid shell, and the assigned dimensions of the coolant passage are ensured.

252

The equivalent diameter for the slot channel according to formula

(4.194)

а Here a decrease in the volume of the channels owing to forgings or Separate inserts will be disregarded,

A common deficiency of chambers having a slot coclant passage is, first of all, the insufficient ability for heat removal of high heat flow appearing at high pressures in the chamber when using high-calorie fuels, etc. The reason for such a relatively low (under conditions of operation of a contemporary ZhRD) effectiveness of heat removal in slot ducts is the insufficiently developed surface of heat exchange on the part of the coolant and the difficulty in the fulfillment of the internal wall with widely spaced fastenings thinner than 1,5-2 mm, which also limits the heat removal, Furthermore, the rigidity of the chamber w‘th slot coolant passage even with Separate fastenings is insufficient,

Coolant Passages with Ribbing

The best heat removal and also the possibility of decreasing the thickness of the inner shell of the chamber 1s provided by ribbing of the inner shell. Furthermore, ribs usually serve also for the fastening of inner and outer shells, The obtained braced shell with frequent connections possesses great rigidity and strength,

Examples of such a construction are coolant passages with longitudinal or screw ribs and with corrugations shown on Fig. 4,26d, e, and f, In the case of providing the full contact of the insert with the shell, the effect of ribbing can take place and with the installation of inserts (see Fig, 4.25b), although the effect of it 18 negligible. An increase in heat transfer from the wall to the coolant owing to the effect of ribbing also takes place and in tubular chambers, .

253

Figure 4.264 gives a diagram of a coolant passage with longitudinal ribs, The longitudinal or screw ribs with a thickness of 121.5 mm are usually obtained by means of milling of the channels along the generatrix of the chamber of the engine or along the screw curve. In the ribbed coolant passage the heat exchange is improved not only because of the increase in cooling surface, but also owing to the possibility of the fulfillment of a thinner inner shell with Son Of the order of 1 mm,

The equivalent diameter for a channel with longitudinal ribs is equal to

в А МА 20 _ . 20+) Sons On Fig. 4.2e a coolant passage with screw channels is shown. The screw channel can be both single-filar and multifilar.

The main advantage of the screw coolant passage consists in the increase in motion rate of the coolant along the passage, which permits increasing heat removal, Thus on the ZhRD "Walter"* the motion of the coolant along the helix (twist) was provided by a screw channel on the external shell. Obviously, in this case an increase in heat removal occurred only because of the increase in speed of the coolant and not owing to the effect of ribbing.

Figure 4.27 shows a cross section of a nozzle of a chamber of the engine ORM-45 where a twist of the fluid was created by a screw thread carried out alcng the inner shell on the part of the depth of the coolant passage.

: АХ

<<< AS |

Fig. 4.27. Coolant passa

age of the possie part of the engine ORM-45,

254

With the noted merits of screw channels, the deficiencies of such ducts are the great laboriousness of their making and the considerable increase in flow friction of the duct,

Therefore, usually screw channels are made not along the entire length of the combustion chamber and nozzle but only on those sections where it is necessary to provide the greatest heat removal, most frequently in the throat, With this to decrease the laboriousness, the screw channel is frequently made simply by means of installing of spiraling various kinds of inserts or wire,

The equivalent diameter for the coolant passage with screw channels will obviously be determined just as for the straight channel;

Figure 4.26f shows a coolent passage in which the channels are formed by an insert of thin band [12]. For simplicity we will call such a duct a coolant passage with corrugations, In the obtained shell corrugations simultaneously play the role of ribs, which increase heat removal, and fastenings, which increase the rigidity and strength of walls of the chamber. With a quite high frequency of corrugations, it is possible to use a thin inner shell, which also improves the heat exchange.

The equivalent diameter dp, for a duct with corrugations is

calculated by formula (4.194) in reference to the geometry of the obtained channel,

Approximately dy for the coolant passage with corrugations (Fig. 4.28) can be determined by expression

— де (4.200)

where ар ~ distance between the ribs along the center line of the "useful cross section" where

255

eh a 29° antic

——_— —-*

0g + 2s, = м fay Ban +) oa. (4.204)

where Zp =— number of ribs of corrugations, In practice, with a sufficient degree of accuracy, it is possible to

consider Bon а. ,

Fig. 4.28. Determination of d, for a coolant

passage with corrugations.

Tubular Chambers

In recent years there has been widespread use of the so-called tubular chambers made of a set of tubes soldered to each other (Fig. 4.29). The tubes can be of round, oval or rectangular section and are located both along the generatrix of the chamber of the engine and also along the spiral, Tubular chambers with spiral winding (Fig. 4.30) have not received widespread use, since along with complexity of their manufacture the flow friction of the coolant passage of such a chamber is very high. Furthermore, with the spiral winding of pipes it is difficult of provide a smooth contour of the inner surface along the longitudinal generatrix of the chamber and nozzle. Therefore, for tubular chambers the longitudinal location of tubes is characteristic.

The most widespread form of the section of tubes is the rectangular or trapezoidal form with rounded angles, There are used both copper or aluminum tubes and tubes made from heat-resistant alloys. The thickness of the tubes, depending upon parameters of operation of a ZhRD and the material, lies within 0.3-1.0 ша, Usually 250-350 tubes are placed along the perimeter of the combustion chamber, To increase the strength of the tubular chamber along its length are

256

sees < —=-

placed several force rings — bands, and on the most stressed sections of the chamber over the tubes frequently there is put & solid lip.

ИЕ 0550 0600 8888

Fig. 4.29. Diagram of tubular chambers; 4 — rectangular tubes; 2 — tubes with ribbing; 3 — round tubes; 4 — two-row distribution of tubes; 5 — spiral tubes; 6 — duct of a profiles; 7 — force lips or winding; — U-shaped profiles; 9 — places of soldering.

` и“ 71) а ны А А

; о a ll

Я

Fig. 4.30. Chamber of an engine from spirally wound tubes,

To increase the strength of the shell of the chamber and nozzle, also the distribution of tubes in two rows, one above the other is possible (see Fig. 4.29a).

The basic advantage of tubular chambers, as compared to other designs of the coolant passage, is the gain in weight. Furthermore, tubes maintain high pressures in the duct and provide good heat removal by the coolant both owing to tne small thickness of the walls

257

and due to the effect of ribbing.

The difficulty of the making of chambers with the “ongitudinal distribution of tubes consists in the fact that along the length of the generatrix the tube should have a variable section (Fig. 4.31). The profile of tubes for chambers having a nozzle with a high expansion ratio is changed especially considerably.

ae

‘ а в _ заб =a 1-1 +

oa @2 ox-,

Fig. 4.31. Change in section of tubes along the length of the generatrix.

с с

The installation of a different quantity of tubes along the length of the chamber of the engine, instead of a change in the profile (Fig. 4.32), is very difficult constructively and is used comparatively rarely.

Fig. 4.32. Tubular chamber with a variable number of tubes,

258

ЕАН

эм о "А�"Х".

| a

A variety of tubular Chambers is chambers comprised of U-shaped profiles (see Fig, 4,29b) over which passes a cooling medium, The profiles are gathered in the form of a Chamber, are soldered along the generatrix and are covered on the outside by a force Winding.

Feed of the Coolant to the Coolant Passuge

The uniformity of feed of the coolant along the perimeter of the coolant passage is very important, since otherwise there can

appear the danger of burnout on sections where the flow rate of the coolant will be less nominal,

To ensure the uniformity of the coolant feed, at the entrance into the coolant passage combined collectors are installed (Fig, 4.33).

Fig. 4.33. Diagrams of coolant feed: 8, b) slot coolant passage; c, d) ribbed Coolant passage; 4 — corrugations; 2 ~— ribs,

In the engine of the A-4 rocket (see Fig, of the coolant feed is ensured by feeding it to equally located tubes, which complicated and rendered heavier the

construction of ‘the whole installation, Subsequent experience showed that one to two feeder tubes is sufficient,

5.4) the uniformity &@ collector on six

259

a Е

The selection of the place of location and form of the section cz the collector is determined by design-technological coneiderations.

The entrance collector is usually set in the nozzle part of the engine, Thus, on Fig. 4,33a, b, and c the collector is set directly near the section of the nozzle, If cooling conditions are allowed, then for a decrease in the overall dimensions of the cross section of the engine and weight of the construction the collector is displaced in the direction of the throat,

The coolant proceeds from the collector into the passage either through slot (Fig. 4.33a) or through holes made in the external shell (Fig. 4.33b). In passages with longitudinal ribs or corrugations the location and dimensions of the holes should provide the entering of the coolant into all channels of the passage (Fig. 4.33c and а).

In tubular chambers it is frequently expedient to locate the entrance collector directly near the head of the chamber (see Fig. 5.3). Here the coolant from the collector enters into every second tube, passes through it up to the end of the cooled part of the nozzle and, returning back through a tube located in series, proceeds directly into the head. Such a method of separation of the coolant is frequently called "two-way" or "in two movements," With such separation of the coolant hydraulic losses in the coolant passage are somewhat increased, but then there is no need in a well-developed section of the collector in the nozzle part and the length the main Supply lines of the cooling component decreases, which leads to a lowering of the weight.

With the location of the tubes shown on Fig. 4.32, the motion

of the coolant through every.other one of the tubes is used only

at the entrance part of the nozzle. In the most thermally stressed, critical section the coolant moves through all tubes in a direction toward the head of the chamber, Such a separation is sometimes — called "sesquipass," With the distribution of the tubes in two rows (see Fig. 4.29, 4), obviously, it is convenient to use the second row of tubes for a reverse motion of the coolant so that the

260

mate

et

Se es tS

Component moves into the inner row of tubes through all the tubes in one direction,

4.11. Calculation of Heat Transfer in 4 ed Coolant Passage

Coefficient of Ribbing

In the presence of longitudinal ribs, corrugations or soldered inserts the heat transfer from the wall of the chamber of the engine to the liquid coolant increases because of the effect of ribbing of the surface, It is convenient to consider such a change in heat transfer as the introduction of concepts of the coefficient of the effectiveness of ribbing Np and effective ccefficient of heat

transfer taking into account ribbing Oy sp? which are connected by the dependence

=e. (4,202)

By knowing Np and Ae. for the assigned ribbed coolant passage, further calculations of cooling will be made as for the usual smooth surface, using the coefficient “ж.р instead of Ч.

Let us find the expressions for determining the coefficient of effective ribbing у (75]. The calculation diagram will be presented in the form of a flat wall with ribs (Fig. 4.34), Let us assume also the following: the coefficient of heat transfer Gy 1s constant all along the perimeter of the coolant passage; the temperature of the coolant Ty.or @long the section of the passage is identical; the tempere- ture of the rib at the base; thermal conduction of the ribbed wall A does not depend on the temperature and is equal to some mean value,

Every rib will be examined as a rod of finite length L = охл< heat transfer from the ends will be disregarded, Then, inasmuch as the temperature at the base of the rod is equal to Ти. от, the temperature distribution over the pitch of the rib and heat flow

through the base of the rib Q will be determined by these expressions [84]:

261 Е

—— en И

tat, chin- —)}-, ch mtg, г Qu= —lfmithm-s,...

Here 4@Tfan—Ta; (=T)—Te; T. — temperature of the surface of the rib, which is variable along its length; x — distance from the base of the rib; 4 — thermal conduction of the material of the ribbed wall;

f and Il — area and perimeter of the cross section of the rib,

Ll �<< АА�< НЙ be SAA VAS АИ Е , 77 ЛЯ, | Fig. 4.34, Calculation diagram for determining

р РИ Г: ‚я УРА. AP ro } A в „ р j Г МР: Oe ee | Wy si MMe LAL LLL a

In coolant passages of a ZnRD the thickness of rib b is considerably less than its length 1. Therefore,

п _ 2+5 _2 7 b&b =-,

whence

"-у>>. ; mo The quantity of heat Q returned to the liquid from the ribbed

wall on the section equal to the spacing of the ribbing (a + b) is determined by expression

ЧАИ и Te)+ MIM Tay—T Jt hye Pan T0)(0+)/ Вь В, where a — transit between the ribs,

262

— — И

Since the surface area of the gas wall

Frey (a+ by,

the specific heat flow Qs» referred to this area will be

й тв (Ти Та atm

(4,203)

Designating

| Ищи (4,204 )

ana

. Shere, (4.205)

expression (4,203) can be copied in the form

93=O,0, (Ta er—-T gs (1.206 ) | where TEE sO. (4.207)

of effectiveness of ribbing. For the convenience of calculation

| Expression (4,207) is the formula for determining the coefficient Fig. 4.35 gives the dependence of f(€) on €,5

iQ), 10

Fig. 4.35, Graph of the dependence of Р(Ё) on ЕЁ.

263

ть тиваидивчирьло, «отровиовимитниьчулвчо позомтуонвьнтоночоечанияноси с ыпуананицы

м а ВИС И Орви

Since according to expression (4,202) Cyn) = Og, ps formula (4,206) can be rewritten in the form

me, (Tai Tad (4,208)

1.е., we obtain formula for the determination of ys which is = similar to formula (4,191) with the difference being that instead of ay enters effective coefficient of heat transfer taking into

account the ribbing Чж.р.

Effectiveness and Optimum Geometry of Ribbing

By analyzing expression (4,207), taking into account equality (4.204), we see that the effectiveness of ribbing at the fixed value dy depends on the thermal conduction of the material \ and geometry of the ribbing. {

With an increase in A € decreases, and f(€) and "р increase,

The reason for the growth in p is the fact that with an increase | in dX thermal resistance of the rib becomes small as compared to the thermal resistance of the heat teansfer to the coolant, and the temperature of lateral surfaces of the rib is compared with the } temperature of the base, owing to which the heat exchange is improved. In the limit when €~ 0, f(@) ~1 and according to equation (4,207)

will be equal to the ratio of the area of the ribbed surface to the smooth surface, i.e., the geometric coefficient of ribbing np instead of Np always gives oversized values of On, p? and only in the case of very high thermal conduction of the material will the values of Тр and Тр be close,

For clarity Fig. 4.36 gives calculation graphs of the change | in temperature of ribbed steel and bronze walls taking into account й Np» taking into account only Пр and neglecting the ribbing. Since for a steel wall the thermal conduction » is comparatively low, then the value of @ is great (usually more then two) and Np noticeably differs from Tp Therefore, the calculation taking into account only Пр leads to decreased values of the temperature of the wall, For a bronze wall for which @ is small and values т and Tp are

264

Similar, results of calculations with

or with Np @re also similar,

From the graphs it also is clear that the calculation of cooling without taking into account ribbing gives in both cases sharply

different results; thus, with ribbing calculation must always be conducted by taking ribbing into account,

Fig. 4.356. Calculation changes in the temperature of the wall with calcula- tion taking into account and neglecting ribbing:

1 — neglecting ribbing;

2 — taking into account Пр;

2 — taking into account only Пр.

The geometry of ribbing (i.e., the relationship between a, b, and 5oxq) also affects its effectiveness.® With an unsuccessfully р Selected geometry of ribbing, instead of the intensification of

heat transfer its impairment can be obtained, b > 20oynf(€) in equation (4.207)

Actually, at value | < 1, Т.е,, heat transfer from

the ribbed surface will be worse than that from a smooth surface,

The boundary of the rational application of ribbing is determined

by the relation

Meus (8) = 6,

or, taking into account equation (4.205),

accthm’

where

„-И=

On Fig. 4.37 by formula (4.209) there is plotted the region of the rational application of ribbing (shaded)

Soxn/d, from which it is clear that for quite high ribs (59y,/b > 2)

= bors а

6

(4,209)

(4.210)

depending upon m' and

265 я

it i a ee

there will be Np > 1 under the condition

(mPa Se <1, (1.211)

Fig. 4.37. Region of the rational application of ribbing.

. +

Let us find the dependence between the width of channel a, thickness of rib b and coefficient of ribbing Np at a fixed depth of the rib бохле TO simplify the analysis, at first let us examine the particular case & > 2. Such values of € are characteristic with steel corrugations of ribs (A = 20-50 kcal/m h-deg) (23.3-58 W/m-deg). Here, as one can see from the graphs of Fig. 4.35, with a sufficient degree of accuracy it is possible to consider f(@) = 1/@, Then, substituting f(&) = 1/€ into equation (4.207) and introducing m! according to expression (4,210), we will obtain

+ «+

yo. (4,212)

Let us introduce designation z= ‘, nny deat Then expression (4,212) can be written in the form УГ.

1+2 4 —— . (4,213)

From equality (4.213) it is clear that when z = ции? there will be Ny = 1, With an increase in 2, 1.е., with an increase in the relative thickness of the rib b/a, we obtain Пр <1, 1.е., the ribbing becomes irrational, With a decrease in z value "p >1, but when z = 0 again we obtain р = 1 (which corresponds to a smooth wall without ribbing). Consequently, in the interval

{

O<rct

266

there is a maximum Ty (Fig. 4.38), Let us find its optimum value

an from the condition 7 70:

Fig. 4.38, Character of the dependence Np оп 2

Taking the derivative, we will obtain

2+*У2-1=0, whence Zo] — 0,5 (YI 4—3), (4.214)

From formula (4,214) it 1s clear that always Zopt < 1, 1.е., а]маув b <a,

The width of channel a is usually determined by technological considerations and the magnitude of permissibl According to, equation (4.212) (4.214)

e flow friction, » With the observance of condition the effectiveness of ribbing will be higher the less a is,

Substituting the value Zont from formula (4.214) into expression

(4.213), we will obtain the maximum value of the coefficient of the effectiveness ribbing for the particular case (€ > 2):

€ . ® 1 1 hee у — ле. (4.215) 267

ie |

a и = №.

Sa a

зе

In general the effectiveness of ribbing is also affected by the depth of the rib бохл. Let us introduce designation 6B = .69x7/e. Then, considering that z = b/a, formula (4.207) can be thus convert:

У: & 14+2—th у. (4.216 )

Equating derivative dn, /az to zero we will obtain the following

equaticn for the determination of Zopt? i 1—2 Ур в aL 1.24 Vitis Е “7st se. РЕГ)

Equation (4.217) was solved оп а computer; the obtained values of Zopt Were then put into equality (4.216), and the value Пр шах Was determined, Final results of the calculations are given on Fig. 4.39 [75]. For convenience the reverse quantity Пр БК is plotted along

the axis of the ordinates,

ae

a2

0 | Fig. 4.39. Determination of Tp

ae opt’

268

The order of determining the geometry of ribbing will be examined with an example,

Example of the calculation of ribbing

Determine the number of ribs n, thickness of the rib b and effective coefficient of heat transfer “а. р for the cylindrical part of the combustion chamber.

Given: diameter of the combustion chamber, including the thickness of the internal wall D, = 250 ша; width of the channel a= 2mm; depth of the rib Soy, = 2.8 mm; thermal conduction of the ribbed wall A’ = 250 kcal/m h-deg (291 W/m-deg); the coefficient of heat transfer from the wall to the liquid for a smooth slot channel a! = 1.210" keal/nchedeg (1.395+10" W/n®-deg).

Solution, As yet the number of ribs is still not determined, and in the first approximation for calculation we take a, equal to the value obtained for @ smooth slot (at).

We deternine:

Prom Fig. 4.39 when B = 1,4 and И. = 0.438, ме 210

чела,

whence i =0,91-O=6.000 Such thickness of rib corresponds to the nuaber of ribs da SEE ow, 269

The coefficient of heat transfer G, for a ribbed channel, obviously, will differ from @j, of a smooth channel, since in connection ° with the obstruction of the passage by ribs the speed of the coolant Woxg Will increase,

Tn accordance with equation (4,196) it is approximately possible to assume

и

let us determine the speed of the coolant through the passage blocked by the ribs Мохл.р:

7 2,1¢-250.9.8 = —_ qa Ome FT 4-00-2.0— 9.190.030 Um Men:

whence

--< Е. №.1.96* 1.4. микса ав. deg

2B -/EBE 0

Prom Pig. 4.39 when Ba 1,4 and xl! = 0,477

ели whence Г ТР eres Gant. The number of ribs a

р The mmber of ribs differs little from the number of ribs obtained

in the first approximation. Therefore, we finally take b = 0.5 m we % max * 2.43. The effective coefficient of heat radiation

ато

from the wall to the coolant taking into account the ribbing is equal to

Sm 9 Sep nen = 142+ 104-2, 48 om 3,55-104 Ксан ‘deg

(лено тая)

Heat Transfer in Tubular Chambers

The expressions obtained above for the calculation of heat transfer from the ribbed surface can be used with approximate Calculations of heat exchange in tubular chambers, Here the calculation diagram of the heat exchange in the tubular chamber can be represented in the form shown on Fig, 4,40

чи УХ ЕР оо / 7

N tok Fig. 4.40. Calculation

ZS у А diagram for the determin- К Их. ation of Чи, тр.

Let us consider that the heat supply occurs only along the internal surface,

The quantity of heat tapped by the external frame or covering will be disregarded, and we will consider it equal to zero. Thus, all the proceeding heat is removed to the coolant through the surface COO'C, Having developed the upper wall of the tube (section 0'C), we will obtain a conditional diagram for the calculation of heat exchange ‘in the tubular Chamber, With this the depth of the rib Soxn is equal to the length of the section 00'С. The thickness of the rib is equal to doubled thickness of the tube b « 26.

The width of the channel a is considered equal to the internal width of the tube on the given section,

271

If one were to take with respect to dy, Ty, Ти. от and » the same assumptions as in the calculation of heat exchange in the ribbed coolant passage, then it is obvious that with the designations accepted in our conditional diagram all the formulas (4,202-4,208) obtained above will remain correct, Consequently, for the tubular

chamber

Carp E53 (4,218)

Gay op (Tver ah р (4 e 219)

where Om.gp — coefficient of heat transfer from the wall to the coolant for the tubular chamber; ay — coefficient of heat transfer for a tubular channel obtained by formula (4.192); т) - coefficient of ribbing determined in designations accepted on Fig. 4.40 according to formula (4.207).

Just as in case of the ribbed coolant passage, there can be formulated the problem on the determination of optimum dimensions of the tube. However, its solution is greatly hampered by a large number of factors affecting the relationships between dimensions of the tubes, The particular question of the determination of optimum thickness of the tube, depending upon conditions of heat exchange and strength properties of the material is examined in work [13]. The problem of the distribution of temperatures in the wall of the tubular chamber is examined in work [87].

4.12, Calculation of Cooling of a ZhRD Order of the Calculation of Cooling

In the designing of a cooling system of a ZhRD, at first there is determined the construction of the coolant passage, method of cooling and basic dimensions of the passage, and then by calculation means it is checked whether there is provided cooling of the walls of the engine. The checking calculation of the cooling of the chamber of a ZhRD is conducted in the following order.

272

Выше:

ОЧ НИИ

1. We divide the combustion chamber and nozzle lengthwise into separate sections, Usually in the nozzle part we take 12-20 and in the combustion chamber 41-4 sections, depending upon their Гоги. In certain cases for obtaining specified data, we divide into separate sections places of the joint of fastenings (corrugations, ribs, etc.), and also sections having a specific form different from the form of the whole coolant passege. After the division with respect to each section, its geometric parameters necessary for further calculations are determined,

2. We assign the temperature of the gas wall Тр.ст constant in length and determined the values of convection heat flows Чх for every section, The order of the determination of dy 18 given in section 4.6, If there are data on the distribution of Чк for the base engine, then it is not possible to conduct a special calculation of Чк.

3. We assign the distribution of Тр. ст along the length of the chamber of the engine, and by formulas of conversion (see section: 4,7) we determine the distribution Чк. With this as the initial values it is possible to take following values of Тр.ст: in the throat of the nozzle 1000-1300°K for heat-resistant steel, 700-900°K for standard construction steels and 500-700°K for walls of copper or its alloys; at the outlet of the nozzle (depending upon the expansion ratio of the nozzle) 400-700°K for steel walls and 300-600°K for copper, In the combustion chamber and at the entrance into

the nozzle T, 4” 18 20-40% lower than the temperature of the walls in the throat,

If there are no preliminary considerations about the character of the distribution of Tp on: Slong the length (data of similar constructions, etc.), then in the first approximation the distribution of intermediate values of Тр.ст can be considered linear, However, if values of Tp.cr for similar designs of a ZhRD are known, it is

more expedient than for the first approximation to take these values,

273

ц, We determine the distribution of radiant heat flows q, along the length of the chamber and nozzle (see section 4.8),

5. We determine the total specific heat flow into walls of the chamber of the engine (section 4,1):

Q=6.+%-

6. We check the sufficiency of flow rate of the coolant for the removal of heat proceeding into walls of the chamber of the engine, With the steady state of cooling, all heat entering into walls of the chamber of the engine goes for heating the coolant. The equation of heat balance

Vitel lees (T uaF) (4.220)

where az, and oF, - total heat flow and surface area of the wall of

the i-th section of the chamber of the engine; Goy; — flow rate of the coolant; Tex and T sax — temperatures of input and oucput of the coolant; Cop — average heat capacity of the coolant taken at a temperature of

тат'�.

From equality (4.220) we determine

rer, (4,222)

The temperature of the input of the coolant T,, is considered equal to the greatest possible ambient temperature at which operation of the ZhRD (for example, +90°C) is assumed, The adequacy of the quantity of coolant is determined by the condition T,,, ¢ 1,, mere 7, — doiling point of the coolant at the outlet pressure froa the coolant passage. If Tom is greater than T,, then it is necessary either to cool by two components according to the diagram of Pig. 4.8 or to decrease the total heat flow into the wall q, by means of amplification of the internal cooling.

74

7. We determine heating of the coolant and average temperature of the coolant on every section. With uhe known temperature of the entrance of the coolant the heating of it 1s determined from the equation of heat balance on section

ah CQ aa (Tne.t Trad (1.222)

where C, — average heat capacity of the coolant on the i-th section, Hence

Tanne +T ы | (4,223)

The temperature of the coolant on section T

mi is defined as an average:

Tyas, (4.224 )

8, We determine the coefficient of heat transfer from the wall to the liquid on each section, (The order of the determination of the coefficient of heat transfer Чи» Seyp » OF Oy en for different forms of the coolant passage is given in sections 4.9 and 4,44.)

9. We determine the temperature of the liquid wall for each section, According to equation (4.191)

тт. (1.225)

where a, — coefficient of heat transfer from the wall to the liquid, which is defined for the given form of coolant passage, i.e., Gao

Sg.pr. OF Gy gn

10. We determine the temperature of the gas wall Тр. ст.

According to the heat-conduct/on equation

Фи Ты

275

whence

where Soy — thickness of the internal wall of the combustion

Tee otin Та,

(4.226)

chamber and nozzle 3 4 — thermal conduction of the material of the wall at the average temperature of the wall

+ Ти, Томе Ты:

Values of thermal conduction for different metals are shown in

Fig. 4.44,

№

Fig, 4,44. Thermal conduction of metals: 1 — silver, 99.9%; 2 — copper, 99.9%; 3 ~ molybdenum; 4 — tungsten; 5 — niobium; 6 —

‘nickel; 7, 8, 9 = sonstruction

steel; 10, 11 — stainless steels; 12 — titanium,

276

11. We check the coincidence of the assigned temperature of | the gas wall with 1, .,, obtained according to formula (4,226), 7

If the divergence of values Тр, ст is more than 5%, we assign the new distribution Th, ст intermediate between that assigned in. the first approximation and the one obtained (but closest to the latter), and the calculation starting from No. 3 is conducted again, If value ®. ст ОП some section exceeds the maximum permissible for & given material, it is necessary either to decrease the thickness of the wall or to improve heat removal from the wall to the liguid, If this is impossible, it is necessary to decrease the heat flows Gy Owing to the intensification of internal cooling,

Example «f the calculation of cooling

Produce a checking calculation of the cooling of the chamber of an engine operating on components (80% HNO, + 20% N50, ) and Tonka 250,

Givens pressure in the combustion chamber ро = 65 kef/om@ (6.37 MN Ane )3 temperature in the combustion cheah ev То = 3070°K; flow rates of components 6; = 48 kg/s; te ак diameter of the combustion chamber D, = 240 mm; throat diameter Dup = 122 mm; diameter of outlet section D, = 439 mm; length of the cylindrical

part of the combustion chamber Ly = 276 mm,

The contour of the combustion chamber and profiled nozzle is shown in Fig. 4,42, The external cooling is by fuel; at the internal ; wall of the chamber of the engine by the corresponding location of . injectors on the head a boundary layer is created with the relationship | of component yo, = 1.9433; the thickness of the »oundary layer is equal to the spacing between injectors H = 145 mm, The coolant passage is made with corrugations; the number of ribs of the corrugation — 243 1t 18 variable in length and on the sections is accepted as the following:

277

nara en

The height of the clearance of the coolant passage охл and

thickness of the inner shell Sop and corrugations b are constant and

The inner shell is made of stainless Steel and the corrugations — from construction steel, .

a я

$ Е : } Ее

\,42, le of the calculation of ain

278

Е 3

|:

From the thermodynamic calculation the partial pressures necessary for calculation of Qy are also know: Piro = 22.4 kgt/cn® (2.2 М/И 2);

Poo, = 7-7 kef/en® (0.75 Мин?)

Solution. The calculation of cooling 1s produced in the order Stated above; all data of the calculation are given in Table 4.4,

1. We divide the combustion chamber and nozzle in length into 14 sections, as was shown on Fig. 4.42,

For simplification of the calculation we will not separate into separate sections the places of the Joint of corrugations, From the drawing of contour we determine for each section the necessary (for further calculation) mean values of the internal diameter D, the cosine of the angle of inclination of contour cos Y, the distance from the effective flame front #, and the length of the section along the axis 4x,, and we calculate the dimensionless values B and 5х, (lines 1-6, Table 4.4),

2. We assign the temperature of the gas wall Тс? = 700°K constant in length, and in the order discussed in section 4.6 we determine for each section the values of convection heat flow 95700 (lines 7-17), The values necessary with calculation Тефоо›

Ви’ бр.вф.ср 816 Hagoo for our fuel when toe = 1.983 are determined from graphs of Fig. 4,16;

a = Fm вв

“eee

We calculate value Reo from formula (4.159): Reo © 49h .a0? (if for some fuel there are no auxiliary values the show can be calculated in the order given in 4,6),

279

3. We acsign the distribution Tp.o¢ 1 length, and according to formulas of conversion we determine the real values of convection

heat flow q» (lines 18-20). A change in Gg in length of the chamber and nogsle is shown in Pig. %.42,

\. We determine the radiant heat flows 92, We find Qa." 7 the combustion chamber, neglecting the boundary layer

Since Lok -B-1.u, the length of the path of the rey will be equal tO sme Rodi -09000is a

Then

Uae Qo 2.4.0.0 8.8 м 7.79 WH 21.3.

We find the degree of blackness of the gas in the combustion chamber ¢p. According to Fig. 4.21 when T, © 3070°K and Puc! © 3-5, the degree of blackness of H,0 when Ро © 1 will be

agree”,

Determining from the graph on Pig. 4.22 the exponent

erl+ins ago

According to formule (4.187) or the graphs of Pig. 4.23 we find *n,0 > бебтее of blackness of HO with the parameters of operation

of the chaaber:

tae? F—tag ol f-Aiey tua

Knowing “н.о and ‘co, according to formule (%.185) we find the the degree of blackness of products of combustion in the chamber:

formule (8.168) determine the effective Gegree of blackness of the well:

denen © teal! + £1 — gel — afm 0.010 + 61 — 0,04 0.009» 0.88. Since walls of the chamber have external cooling, we will 2isregard

the second term in formula (8.164) and determine the Magnitude of radiant heat flow in the combustion chamber Ча. п according to (4.185):

чьим (В олово вл (ль wawhe.nw wel),

We determine the value of radiant heat flow in the combustion chamber taking into account the boundary layer nce: The coefficient @, considering the decrease in intensity of radiation according to formula (4.390) is equal to

eA ae ate пра. Then according to expression (4.189)

foam = OMincetns > 0.4-4.5.4.5.1.0. 26.44 № mesa os w?.)

We determine the distribution Gg along the length of the combustion

chamber: near the head Cones Caner = OM cam = OM- 1 —0.9-4,05-19 000-10 beatae @-IW we?

в hroat .

Grape Clans 08-415. = 0,4. neet ae нии. АЛЬ

201

ip the secticn where Deo 2.5

Trecing the graph of distribution 4; along the length of the chamber of the engine (see Fig. 4.52), we determine the value of Ча Гог every section (line 21).

5. We determine the total heat flows

Co Ce + Ge-

Values 9; аге given in line 22, The change in Qs along the length of the chamber and nozzle is shown on Pig. 4.42,

6. We check the adequacy of the quantity of the coolant. We consider the teaperature of the input equal to 450°C (323°K). Then according formula 4,221 the temperature of the coolant outflow

os т +7 при + ес.

1.0.5 Tye < Ty

7. We determine heating of the coolant and average temperature of it on each section (line 23-27).

8. We determine the coefficient of heat transfer from the wall to the liquid, taking into account ribbing Qg.p (lines 26-40), The complex 26 and the coefficient of thermal conduction of corrugations Ap were determined on the assumption that the temperature of the Liquid wall is constant and equal to 200°C (473°K). such an assumption does not considerably affect the value 26 and € and, respectively, вп 284 аз. р, since in the region of the change in т. of, (line 42), Pre og ond "all the wore во В, (Proportional to (Prg.gg) ">, and also ¢, proportional to (Хэ) 0.3, are changed very insignificantly, Thue, according to the auxiliary graphs of Fig. 4.25 or formulas (5.295) and (4.196) 28 @ const @ 0,088, and according to the graph of Pig. 4.814 A, @ 60 keal/a hedeg on all the Sections, In the given

example the impairment of value » OWing to the increase in thermal resistance at places of soldering the corrugation, is not considered,

9. We determine the temperature of the liquid wall by formula (4.223) (lines 41-12),

10. We determine the temperature of the gas wall obtained under the assigned conditions of cooling (lines 43-46),

11. A comparison of the obtained Theor (line 46) and assigned (line 18) shows that the divergence between them does not exceed 5%, ап@ therefore we consider the calculation final,

4,13, Certain Special Cases of = Coottng ЗА

We will examine the special features of radiation cooling of a@ ZnRD, cooling by low-boiling components, and cooling of a ZhRD having a nozzle with a central body,

Radiation Cooling

Above in section 4.2, radiation cooling is examined as one of the methods of cooling a ZhRD. Such cooling can be expedient for @ ZhRD with a large nozzle expansion ratio and a prolonged operating time (for a comparatively short time of operation ablative cooling is more rational). Radiation cooling can be applied in those engines which are set on the external surface of the vehicle, The basic problem in the calculation of radiation cooling is the determination of the value of the equilibrium temperature of the wall, Тот.р.

Determination of Trp during radiation cooling

In the general case the following balance of heat fluxes occurs during radiation cooling (Fig. 4.43):

Gat a™ Fant Fam (4.227)

283

ана

where q and Gg 4re convection and radiant fluxes in the wall of

the engine chamber; Чк.н and In.y 2h@ Convection and radiant fluxes into the environment,

Equation (4.227) 1s also the fundamental equation for determination of the equilibrium temperature of the wall т.

че, 4]

т.р’

Fig. 4.43, For calculation of radiation cooling,

The values of heat fluxes entering into it are determined as follows, The convection heat fluxes from the gases to the wall and from the wall into the environment are equal, respectively,

= 6, (Tig ет} (4 . 228) Фа, (Т.„-Т.), (4.229)

where T = just as in preceding calculations, is determined from equation (4,27) with a component ratio т in the near-wall layer (if there is one) or in the flow core.

The coefficients of convective heat transfer from the gases to the wall and from the wall into the environment, “т апа a, are determined by the known formulas of heat transfer, taken in reference to the given conditions of heat exchange,

In the particular case, when values of Ч» а some given Tn Jon on a certain section of the nozzle are already known from calculations

of cooling of ZhRD, it is possible to determine a, by the forma

щи. (4,230)

284

During further calculations in the first approximation it is possible to consider G, to be independent of the wall temperature,

During engine operation in vacuum conditions, obviously, ay and, accordingly, In.» are equal to zero,

The magnitude of the radiant heat flux Gy 18 determined by equation (4.484) or approximately by equations (4,182) and (4.183) with replacement of Ty by the thermodynamic temperature Treo of the combustion products in the flow core and of Тр.ст by the equilibrium

temperature Тет.р. Неге Gq can be directed both from the gas to the wall and vice versa,

The value of the radiant heat flux into the environment is determined by the equation

чьи (1), (в.234)

where Ест 18 the degree of blackness of the wall material; Tbe.p is the external equilibrium wall temperature,

With a sufficient degree of accuracy it 1s possible to consider Г. м Tes = Ter

Substituting the expanded expressions for heat flux in equation (4.227) and solving it, we shal] determine the values of equilibrium wall temperature Тст.р for 4ifferent sections of the nozgle. r¢ the greatest wall temperature permissible for the given material is known, it is possible to find the section, starting from which the nozzle unit can be made without special cooling.

Example of calculation of equilibrium wall temperature

Determine the equilibrium temperature of the wall of the nozzle

of an ZWRD made from chrome-nickel alloy. The parameters of the Zhrp which determine the heat exchange in a given section are as follows,

295

Given: nossle diameter Do 0.5 mm; effective stagnation teupera- ture in the near-wall layer Tygon = 1800%; coefficient of heat transfer Gy © 250 keal/n“hedeg (291 М/а?. 405); a, = 0 (heat trensfer in vacuum), The thermodynamic teuperature of the flow core 2 , = 1000°R; partial pressures of .0 end 00, are

Pag ОВЛ wajeh, Ро, ОА. му.

Solution. Let us express the values of the heat fluxes entering into equation (&.227) through the equilibrium temperature of the wall Toe.ps for which we will deternine Cog.gp O04 ep entering into expression (4.184), ‘The length of the path of a ray will be determined as for a cylindrical shell of infinite length:

600,090 0946500.68 Фар д 60.40.06 ОД Cen f 9 0184,6~ 0.00,

SE COREE Syn OF Oy,» From the graphs in Pigs. 4,21 and 4.22 at T), = 1000

qa eO; 0014 tg rye.

tag 21 mag gy 1 -0-—040 =6 м, From the graph in Pig. 4.2% ‘co, ° 0.055. We determine ¢, by formula (4,18): ор > gp, — Masten, © OM + 6,0 — 0,67 -O,48 & 6,191.

Considering that there is soot on the internal wall, its degree of blackness is taken as воз = 0.8,

Then by formula (4,186)

The radiant heat flux from the combustion products to the wall will be determined according to equation (4.184), considering tp © Apt

ие БТ ВУИ) С.

The sign of q, (i.¢., the direction of heat flow) will depend on the relationship between T, , and Toe.p°

The convective heat flux is

$=, Pqn—Tang)@ MO —Teag). (4.233)

We will determine the radiant heat flux from the external wall into the environment.

According to Table 4.3 the degree of blackness of the external wall is

te = 6,8.

Considering Tor.p to be identical over the thickness of the wall, we find q, _ by equation (4.251):

и a.m)

Placing the expressions for Qg» Gy, and Ча. и in the equation of balance (4.227), we will obtain

wns) mom понт

or

прин}

It is most convenient to solve the obtained equation graphically, Constructing the dependence of the right and left parts of the equation on (a): we will find the unknown value of the equilibrium temperature of the wall (Fig. 4.44):

Тиз = ВИК.

For appraisal of the magnitudes of the heat fluxes we find Чл, Чи» 274 Ча. и. From equations (4.232), (4.233), апа (1.234)

0 [5] вле воду» А

(minus sign indicates reverse direction of flow (from wall to gases) ]

a 200 (1000 — 4) 1,21-108hn0n}/u% (1,41-100 Wat, ons 0.00 (EEE) 1,1-108 neato (1 20-108 War,

Comparing the obtained results, we see that in this case the value of q, is very small as compared with qs and Qn.y and is equal to approximately 10% of the radiant heat flow into the environment,

‘Pig, 4.44, For the example of determination of equilibrium wall temperature Tor.p*

‚ 288

fe TT

ae

Fig. 4.45, Dependence of equilibrium wall temperature on Tsg00°

Frequently during radiation cooling the fraction of Gy can be even smaller, Therefore during rough calculations of radiation cooling it is possible to disregard radiant heat transfer from combustion products to the wall, If it is also possible to disregard convection into the environment (during operation of ZhRD in vacuum or in a very rarefied medium), then equation (4,227) is simplified and at Thor” TOnep = Тот.�' in expanded Гога, taking into account expressions (4,228) and (4,231), has the form

Tern \4

By solving this equation, using 3 as a parameter, it is possible to construct the graphic dependence of Tor.p on T 9003 this allows immediate estimation of the expected value of Тет.р (in the absence Gy and Чин) (Fig. 4.45),

Analyzing the values entering into formula (4,227) and the graphs in Fig. 4.45, we see that Тет.р also depends to a considerable extent on the degree of blackness of the wall, Ест. The equilibrium temperature of the wall is inversely proportional to the magnitude of Ы Therefore during radiation cooling it is better to leave the external radiating surface, since degree of blackness of a rough surface is greater than that of a polished one. In certain cases it is even expedient to create special grooves on the radiating surface, increasing the apparent degree of blackness of the surface

by 50-100% [86].

289.

РАНА

External Cooling by Low-Bolling Components

| During operation of @ ZhRD on low-boiling components (for example, 0, + H, or Fo + H,) it 18 impossible to organize external cooling of the chamber walls by the liquid component alone, in view of the very low boiling points of both the fuel and the oxidizer, The coolant, upon entering the coolant passage, is quickly turned into vapor and further external cooling of the chamber is then produced by cold gas (vapor). The problem of reliable cooling of the chamber walls of the engine is severly complicated by the following factors.

During the transition of the coolant from the liquid into the gaseous state the coefficient ./ heat transfer from the wall to the gas is less than the a. of the liquid coolant.

The specific volume of the gaseous coolant is considerably greater than that of the liquid, and is strongly increased with a rise in temperature (by two-three times), Therefore it is sometimes necessary to make the cross section area of the coolant passage variable in length so that in every section of the duct the speed of the cooling gas will ensure a value of a, corresponding to the arriving heat flux,

The complexity of solving the problem is aggravated also by the fact that under the conditions which obtain in a ZhRD the cooling of the chamber walls by a low-boiling component occurs in the near- critical and supercritical regions of parameters of state of the coolant. Moreover, in the near-critical regicn a sharp change occurs in the physical properties of the coolant, rendering a strong influence on heat exchange. Thus, for example, the heat capacity of hydrogen in the near-critical region is increased by 5-10 times.

In comparison with other low-boiling components hydrogen has the best cooling properties, since it has high values of heat capacity (approximately 5 times greater than that for water and four times greater than that for oxygen).

290

Possible regions of state of hydrogen during if use a8 a coolant for @ ZhnRD

In the general case the curve of phase equilibrium P, = f(T) has the form depicted on Pig. 4.46.

Abn-~---<28

Pig. 4.46. Curve of phase equilibriua,

The curve of phase equilibrium, OK, has finite length and terminates at critica) point K, corresponding for every Given substance to the fully defined values of temperature Тк and pressure Рк.

At values of pressure and temperature higher than critical the substance is in the region of supercritical parameters of state, In this region it is no longer permissible to talk about the phase transition of a substance, since the substance does not have a liquid and gaseous phase boundary and there is no fundamental distinction between liquid and gas. Although it is difficult to visualize distinctly such a medium, lacking boundaries, it is acceptable to consider that a substance in this region constitutes a conglomerate of groups of molecules with different density, Depending upon the values of pressure and temperature, the substance is in a state closer to gaseous or closer to the liquid, r¢ hydrogen is examined from the point of view of its use a8 a coolant for ZhRD, the diagran which characterizes the state of hydrogen (Pig. 4.47) can be somewhat tentatively divided into the following five regions:

é

1. Region of boiling, 2. Region of the near-critical state,

3. Region of state of hydrogen at a pressure considerably exceeding the critical (№:2.8 kgt/cn*) and at &@ comparatively low

291

temperature (~ up to 150°K).

4, Region of state of hydrogen at supercritical pressure and high temperature (above 150°K).

5. Region of gaseous hydrogen at subcritical pressure. :

At present there are very few recommendations on the calculation { of heat transfer to hydrogen in the boiling, пеаг-сг1$1са1, and supercritical states. For approximate calculations it is possible to use the schemes proposed in works [92] and [97] for determination of аж Гог different hydrogen states,

Supercritical

pressure Subcritical

pressure

Fig. 4.47. Regions of state of hydrogen.

During calculation of cooling by hydrogen for determination of Gy in the first approximation it is possible to use formula (4,192).

Order of calculation of cooling by a low-boiling component

The calculation of cooling by low-boiling components is conducted by the method of successive approximations. For this the combustion chamber and nozzle should be separated along their length into 15-20 sections and every section should be calculated consecutively. It is possible to propose the following order for the calculations.

292

1. Pind the distribution of temperature of the ges wall (ete) Тр ст over the length of the engine chamber and use the methods outlined above to determine the distribution of Чи, Чд› and 9; along the length,

2. Rnowing 9; 0nd T, og. and also the thermal conductivity A and thickness Sop Of the internal wall, determine the temperature of the liquid wall by formule (4.225): Ten@Touq— SEE,

Where the value of Х 16 taken for average temperature of the wal) equal to

a

3. Determine the value of Gq necessary for removal of the total heat flux, By formula (4.191)

о (\.236 )

Take the value of Та @8 average for each section:

7. татьи.

(ax = inlet; suy « outlet /outflow])

\. Determine coolant preheating af, and the temperature of Coolant outflow on each section, According to formula (4.222)

ye SE, (4.237)

where oF, = heat exchange surface area ой а given section; бра > heat Capacity at mean values of pressure and temperature on a given section,

Knowing af..., find

295

5. Pind the coolant pressure Poxa.ax 0m the inlet into the first section.

6. Prom the formas for calculation of GQ, determine the speed of coolant Woy, which ensures a value of ag equal to that obtained by expression (4.236). Mean values of coolant temperature and pressure on a given section are used in the calculations.

7. Mnowing coolant speed Мохд› Getermine the dimensions of the flow section of the coolant passage:

Sani... (4.238)

Wig ess

where the value of Pox, 18 taken with average parameters of state on @ given section,

Since the diameter of the coolant passage over the circumference of the cross section of the engine chamber is know, the determination of охл usually reduces to determination of the height of the coolant passage Sous

8. Determine the loss of pressure 4p, on a given section and the pressure of the coolant on the outlet of the section. The quantity Gp, 18 made up of local losses (4p,), losses on friction (en ), and dims losses of pressure as a result of acceleration of gas а a given section (Ap, ):

The order of determination of yo Sep, and ap, is given below, in section 6.6. It is necessary to note that the magnitude of friction

losses during motion of hydrogen along the passage (2 ep ) is comparatively small, due to the low viscosity of «кн 1

Knowing &s» determine coolant pressure on the outlet of the

‚ sections

Аилля "= Риш — Р (4.240) 294

9. Conducting consecutive calculations for all sections, analyze the acceptability of the obtained results and if necessary Correct or repeat the calculation, If the calues of foxa (OF Soyq) obtained on Separate sections difficult to fulfill in practice, the assigned value of Тр. ст is corrected and the calculation 1s repeated.

Knowing the pressure loss over the length of the entire duct (Zp, » Getermine the pressure Рохл.вых Of the component at the Combustion chamber inlet (Fig, 4.48). If the obteined value of Poxn. BuX differs from the Given value, the calculation 18 repeated with a new value of coolant pressure Рохл.вх ОП %Пе inlet,

Fig. 4.48, Change in Рохл: Ти, and height of

coolant passage tube during cooling by hydrogen,

As can be seen, the calculation of cooling by low-bolling components is considerably complicated not only by difficulties connected with determination of GQ, for different states of the coolant, but also by the fact that, in contrast to cooling by dropping liquids, the calculations must take into account the compressjuility of the component and the influence of pressure losses along the length of the duct on the process of heat exchenge, `

Figure 4,48 shows typical graphs of the change in coolant pressure and temperature and also in the area of the passage section (height of channel 6oy3) along the length of duct during cooling of a can-type (tubular) chamber by hydrogen,

Heat Exchange in Nozzles with a Center Body

One of basic difficulties in using nozzles with a center body in ZhRD is the complexity of organizing the cooling of such a nozzle,

295

These complexities are Caused in the first place by the fact that, in Contrast to the usual ZhRD with a Laval nozzle, the critical section and combustion Chamber of an engine with a nozzle with a center body have annular form,

Due to this Shape the perimeter of the critical section and combustion chamber is sharply increased, Elementary calculations show that for a nozzle with a center body the perimeter of critical Section (place of the most intense heat exchange) is increased by 5-10 times, Furthermore, on one Side of the critical section there is

usually a sharp edge, the cooling of which is in itself a complicated problem,

The surface of an annular combustion Chamber is also considerably larger than the surface of a chamber of the usual shape, which leads

to an increase in the total quantity of heat output on the section of the combustion chamber,

The large perimeter of the critical section of the annular shape “severly hampers the creation of a reliable near-wall layer for internal cooling. The area occupied by the near-wall layer in the critical section grows in proportion to the perimeter of the section and this leads to its obstruction, Especially strong obstruction occurs with large diameters of the center body, when the annular critical section resembles a narrow 810%. In this case in the section in general no place is left for a reliable protective near-wall layer,

Cooling of an engine with a nozzle with a center body is addition- ally hampered by the fact that such engines have two independent coolant passages (see Fig, 2,21) ~ over the contour of the center body and over the external contour, Depending upon the type of nozzle, one of these ducts ensures cooling of the chamber ahead of the critical part and the second provides cooling of both the chamber and the transcritical part of the nozzle. This presence of two coolant passages strongly complicates input and the subsequent . "separation" of coolant along the passsge,

296

Calculation of cooling of nozzles with a center body,

Calculation of cooling for nozzles with a center body can be Conducted in the same order as that for the usual nozzles, We will note only certain peculiarities in using the calculation formlas obtained above for determination of Чк.

During calculation of cooling of round nozzles we selected the the critical throat diameter Dep as the characteristic dimension, which in this case is impossible because the critical section is annular, However, we can select as characteristic some other dimension of a nozzle with a center body — for example, the usually given value of the small diameter d of a toroid or annular chamber (Pig. 4.49).

Effective flame front

Fig. 4.49, For calculation of cooling of a nozzle with a center body,

Then in the expressions for calculation of 2: the dimensionless length of the section X and diameter of the center body B will equal

sme; выс. (4.261)

Since the usual gas-dynamic relationships which determine 6 and Р/Роо ав £(D) are inapplicable for a nozzle with a center body, formula (4,146) also is inapplicable and calculation of Z— must be Conducted by formula (4,141) (with constant Troe):

a= ЗЕ Вах,

where зы. ; Wis the speed of the flow "core" near the chamber

wall, i.e., flow rate along the rarefaction wave incident to a given

297

point of the contour; p is the pressure near the contour at the given point; be and a» are determined by equations (4.127) and (4.129),

We will obtain the expression for R€og (4.159), replacing Dep with the new characteristic dimension d:

= 1107-10 ap 4, 2he чи Zeal V Recs a The formula for determining q, is obtained from formula (4.160), after substitution of the values of o« and w from expressions (4.155) and (4,161) and calcvlation of the influence of the Pr number by formula (4.172) in a form analogous to equality (4.173):

4.=450-1 Вр, —— wen р (4.243) фо

where p is the static pressure near the contour at the given point.

Footnotes

Rules and methods of averaging are analyzed in detail in works

[94], [127].

“The obtaining of equations of the turbulent boundary layer can be found in detail in works [18], [28], [94].

ЗТпе derivation of integrai equations, their solution, and an analysis and obtaining of dependences for determining Чк are conducted on the basis of works of М. М. Iyevlev [77], [78].

*See work [25].

SIt must be noted that due to the accepted assumptions expression (4.207) gives oversized values of Np: Furthermore, with the calculation

for for the coolant passage with soldered corrugations values of % are obtained even more oversized, since the edditional thermal resistance at soldering places of corrugations to the wall is not considered,

ФА detailed analysis of the effect of the eometry of ribbing on 1, is given in the work of V. S. Zarubir (769.

298

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